optimaltime-consistentinvestmentandreinsurance ... · young [7] ﬁrst investigate the proportional...

Research ArticleOptimal Time-Consistent Investment and ReinsuranceStrategy Under Time Delay and Risk Dependent Model

Sheng Li 1 and Yong He2

1School of Economic Mathematics Southwestern University of Finance and Economics Chengdu 611130 China2School of Mathematics Physics and Data Science Chongqing University of Science and Technology Chongqing 401331 China

Correspondence should be addressed to Sheng Li 1180202z1002smailswufeeducn

Received 26 May 2020 Accepted 16 July 2020 Published 28 August 2020

Guest Editor Wenguang Yu

Copyright copy 2020 Sheng Li and Yong He -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper we consider the problem of investment and reinsurance with time delay under the compound Poissonmodel of two-dimensional dependent claims Suppose an insurance company controls the claim risk of two kinds of dependent insurancebusinesses by purchasing proportional reinsurance and invests its wealth in a financial market composed of a risk-free asset and arisk asset -e risk asset price process obeys the geometric Brownian motion By introducing the capital flow related to thehistorical performance of the insurer the wealth process described by stochastic delay differential equation (SDDE) is obtained-e extended HJB equation is obtained by using the stochastic control theory under the framework of game theory Under thereinsurance expected premium principle optimal time-consistent investment and reinsurance strategy and the correspondingvalue function are obtained Finally the influence of model parameters on the optimal strategy is explained by numerical analysis

1 Introduction

Since insurance companies have been allowed to enter thefinancial market for investing risk assets the optimal in-vestment strategy has become an important research topic inrecent years Many literature have studied the maximizationof the utility of the terminal wealth or the minimization ofthe ruin probability of the insurer Browne [1] uses thesurplus process given by the diffusion risk model to study theinvestment problem ofmaximizing the utility of the terminalwealth and minimizing the ruin probability of an enterpriseand obtains the explicit optimal solution Hipp and Plum [2]apply the CramerndashLundberg model to describe the insur-ance surplus process based on the assumption that there isonly one risky asset in the financial market and the time isdiscrete the investment problem is studied Wang et al [3]use martingale approach to study the optimal portfolioselection of insurers under the criteria of mean-variance andconstant absolute risk aversion utility maximization Formore similar literature see Liu and Yang [4] Yang andZhang [5] Wang [3] and Bai and Guo [6]

In addition to market risk the insurer will also considerinsurance risk It is impossible to avoid insurance risk byinvesting in bonds and other assets in the market aloneHowever reinsurance business provides a way for theinsurer to avoid this risk In recent years this approach hasbeen widely concerned Reinsurance business mainlyadopts two different forms of insurance excess-of-lossreinsurance and proportional reinsurance Promislow andYoung [7] first investigate the proportional reinsuranceand investment Bauerle [8] considers proportional rein-surance and investment also and the optimal explicitsolution of the investment-reinsurance problem is obtainedunder the mean-variance criterion Zeng and Li [9] alsostudy proportional reinsurance and obtain the efficientfrontier of the mean-variance under the multidimensionalrisky asset model -e stock price in the above modelgenerally follows the geometric Brownian motion themarket price of stock-related risk is constant but in the realmarket stock price may have other characteristics such asstochastic volatility Liang et al [10] used the Orn-steinndashUhlenbeck process to characterize the instantaneous

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9368346 20 pageshttpsdoiorg10115520209368346

return of stocks and obtained the optimal reinsurance andinvestment strategy Gu et al [11] investigate the excess-of-loss reinsurance-investment problem under the constantelasticity variance (CEV) model

-ere are two deficiencies in the above literature thatdeserve further discussion On the one hand these literatureimplicitly assume that all insurance businesses of insurersare independent of each other so they only study the in-vestment and reinsurance of a single insurance businessHowever in the real insurance market there are often in-terdependencies between insurance businesses For exampleduring the 2019-nCoV medical claims and death claimsoften occur together In order to depict this kind of de-pendency between different insurance businesses the riskdependent model is proposed -e main works in this areaare as follows Yuen et al [12] taking the expected utilitymaximization of the terminal wealth as the criterion con-sidered the optimal proportional reinsurance problem withmultidimensional risk dependence by using the diffusionapproach method For the detailed process of diffusionapproximate to the compound Poisson process see Gandell[13] Liang and Yuen [14] under the principle of variancepremium investigated the optimal proportional reinsuranceof the Poisson model and diffusion approximation modelMing et al [15] derive the explicit expression of the optimalproportional reinsurance under the mean-variance criterionby using stochastic linear quadratic control Considering thecombination of investment and reinsurance Bi et al [16]obtains the optimal investment-reinsurance strategy formean-variance under the diffusion approximation model Biand Chen [17] under the criterion of maximizing the ex-pected utility of terminal wealth arrived at the optimalinvestment and reinsurance strategies On the other handmost of the literature on optimal investment-reinsuranceand other optimal control problems focus on time-delay freecontrolled systems In fact financial markets tend to rely onthe past Chang [18] considers the investment and con-sumption problems related to the return on risk assets andthe historical performance Federico [19] introduces thetime-delay state process by considering the capital inflowoutflow related to performance Peng et al [20] and Yu et al[21] study the optimal dividend policy based on observingthe information of past time points to determine the be-havior of the next moment In fact this is a discrete case oftime delay However the stochastic control problems ofsystems with time-delay state may be infinite-dimensional incontinuous cases hence it is difficult to find the analyticalsolutions Only in some special cases it is finite-dimensionaland the problem has explicit solution Elsanosi et al [22]emptyksendal and Sulem [23] and David [24] provide a theo-retical basis for solving such problems Shen and Zeng [25]first introduced the time delay in the investment and in-surance problem -ey introduced the inflowoutflow ofcapital in the wealth process of insurer and then depicted thewealth process of insurance companies through the sto-chastic delay differential equation (SDDE) After that Li [26]and Lai [27] studied the optimal investment-reinsuranceproblem with time delay under Heston and CEV modelsrespectively

Inspired by the above research this paper combines riskdependence with time delay to consider investment-rein-surance problem -e structure of the rest of this paper is asfollows In Section 2 the financial model framework of thispaper is given assuming that an insurer can invest in a risk-free asset and a risky asset and in the case of two-dimen-sional dependent claim compound Poisson model and theintroduction of the historical performance of the insurancecompany the companyrsquos wealth process with time delay isobtained In Section 3 considering the mean-variancepreference criterion the time-inconsistent optimizationproblem is defined and the extended HJB equation is ob-tained by using the stochastic control theory in theframework of game theory In Section 4 under the principleof reinsurance expected premium the explicit solutions ofoptimal investment and reinsurance strategies and theircorresponding value functions are derived In Section 5 thenumerical calculation process of optimal investment andreinsurance strategies are introduced through numericalexamples and the influence of important model parameterson optimal strategy is analyzed Section 6 concludes thispaper

2 The Model

Suppose that model is based on the probability space(ΩF F tisin[0T]P) of information flow which satisfies thegeneral assumptions of right continuity and completenesswhere T is a finite constant representing the operation cycleof an insurance company and F tisin[0T] is the sum of in-formation available up to time t All stochastic processesinvolved in this paper are assumed to adapt to F tisin[0T]

Suppose an insurer has an insurance portfolio businesswhich is composed of two different insurance businesses suchas medical insurance and death insurance Suppose that therandom variable Yi ige 11113864 1113865 represents the claim amount of thefirst type of insurance business they are independent and havethe same distribution function FY(y) Zi ige 11113864 1113865 represents theclaim amount of the second type of insurance business they areindependent and have the same distribution function FZ(z)We assume that if yle 0 then FY(y) 0 Otherwise0ltFY(y)le 1 And also assume that if zle 0 then FZ(z) 0Otherwise 0ltFZ(z)le 1 In addition their moment gener-ating functions MY(ι) and MZ(ι) exist -e cumulative claimprocess of the two insurance businesses are as follows

C1(t) 1113944

1113957N1(t)

i1Yi

C2(t) 1113944

1113957N2(t)

i1Zi

(1)

where 1113957N1(t) and 1113957N2(t) represent the number of claims forthe first and second categories of insurance business up totime t respectively And suppose Yi ige 11113864 1113865 Zi ige 11113864 1113865

1113957N1(t)1113864 1113865tgt 0 and 1113957N2(t)1113864 1113865tgt 0 are independent of each otherFor different insurance businesses it is assumed that

they are interdependent as follows

2 Mathematical Problems in Engineering

1113957N1(t) N1(t) + N(t)

1113957N1(t) N2(t) + N(t)(2)

where N(t) N1(t) and N2(t) are three independentPoisson processes and the corresponding intensities are λλ1 and λ2 respectively -erefore the total claim amount ofthese two types of the insurance business is

C(t) C1(t) + C2(t) 1113944

N1(t)+N(t)

i1Yi + 1113944

N2(t)+N(t)

i1Zi (3)

Suppose for arbitrary ι isin (0 ζ) E[YeιY] and E(ZeιZ)

exist And for some ζ isin (0infin] there arelimι⟶ζ E[YeιY]⟶infin and limι⟶ζ E[ZeιZ]⟶infin

For convenience of writing we define

a1 ≔ E C1(t)1113858 1113859 λ + λ1( 1113857μ1Y

b1 ≔ Var C21(t)1113960 1113961 λ + λ1( 1113857μ2Y

a2 ≔ E C2(t)1113858 1113859 λ + λ2( 1113857μ1Z

b2 ≔ Var C22(t)1113960 1113961 λ + λ2( 1113857μ2Z

(4)

where μ1Y E[Yi] μ2Y E[Y2i ] μ1Z E[Zi] and

μ2Z E[Z2i ]

Considering the financial market it is assumed thatassets are traded continuously in time interval [0 T] and taxand transaction costs are not considered Suppose the in-surer can invest its wealth in the financial market composedof a risk-free asset and a risky asset -e risk-free asset priceprocess B(t) is

dB(t) rB(t)dt t isin [0 T]

B(0) 11113896 (5)

-e risky asset price process S(t) is as follows

dS(t) S(t) α1dt + σdW(t)1113858 1113859 t isin [0 T]

S(0) s01113896 (6)

where r α(gt r) and σ(gt 0) are constants representing risk-free interest rate drift rate and volatility respectivelyDefine α ≔ α1 minus r

As usual the surplus process from the insurer up to timet is defined as follows

R(t) R0 + ct minus C(t) (7)

where R0 is the initial surplus and c is the premium rate Inaddition it is assumed that insurance companies can con-tinuously reinsurance insurance business in a certain pro-portion to control business risk We denote the retentionratio of categories 1 and 2 insurance business byq1(t) isin [0 1] and q2(t) isin [0 1] When the claim occurs theinsurance company pays q1(t)Yi or q2(t)Zi while the re-insurance company pays (1 minus q1(t))Yi or (1 minus q2(t))Zi Letthe reinsurance rate be δ(q1(t) q2(t)) at time t

LetX(t) denote the wealth process of insurance companiesat time t p1(t) denote the amount of capital invested in therisky asset and thenX(t) minus p1(t) denote the amount of wealthinvested in the risk-free asset -e investment-reinsurance

strategy π(t) ≔ (p1(t) q1(t) q2(t)) will be applied by theinsurer Given an investment-reinsurance strategy π(t) thewealth process Xπ(t) of an insurer satisfies the followingstochastic differential equation

dXπ(t) rX

π(t) + αp1(t) + c minus δ q1(t) q2(t)( 1113857( 11138571113858 1113859dt

+ σp1(t)dW(t) minus q1(t)C1(t) minus q2(t)dC2(t)

(8)

Next we consider the influence of historical perfor-mance on the wealth process Suppose that f(t Xπ(t) minus

Lπ(t) Xπ(t) minus Mπ(t)) represents the inflowoutflow of

capital then the wealth process of insurers with time delay isgiven by the following stochastic delay differential equation(SDDE)

dXπ(t) rX


+ σp1(t)dW(t) minus f t Xπ(t) minus L

π(t) X

π(t) minus M

π(t)1113872 1113873dt

minus q1(t)dC1(t) minus q2(t)dC2(t)

(9)

To make the problem easier to deal with consider alinear capital inflowoutflow function that is

f t Xπ(t) minus L

π(t) X

π(t) minus M

π(t)1113872 1113873

c1 Xπ(t) minus L

π(t)1113872 1113873 + c2 X

π(t) minus M

π(t)( 1113857

c1 Xπ(t) minus

Lπ(t)

11139460

minus he

As

ds⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + c2 Xπ(t) minus M

π(t)( 1113857

c1 + c2( 1113857Xπ(t) minus c1L

π(t) minus c2M

π(t)

(10)

where c1 gt 0 and c2 gt 0 are constants c1 c111139380minus h

eAuduLπ(t) 1113938

0minus h

eAuXπ(t + u)du Lπ(t) Lπ(t)1113938

0minus h

eAudu andMπ(t) Xπ(t minus h) represent the integrated average andpoint by point delay information of wealth process in timeinterval [t minus h t] A(ge 0) and h(ge 0) are given averageparameters and delay parameters respectively Note thatLπ(t) is defined as the weighted average value of wealth

process Xπ(middot) in time interval [t minus h t] and the exponentialdecay factor eAu(u isin [minus h 0]) represents the weight Whenh 1 Xπ(t) minus L

π(t) and Xπ(t) minus Mπ(t) represent the av-

erage gain or loss and absolute gain or loss of wealth ofinsurers in the last operating cycle Because the inflowoutflow of capital is closely related to the past performanceof the wealth process If the past performance is good thecompany will give part of its earnings to shareholders or givebonuses to the management which shows the outflow ofcapital ie fgt 0 At this time Xπ(t)gt L

π(t) and

Xπ(t)gtMπ(t) On the contrary if the past performance ofthe insurance company is not good the company needsadditional financing to achieve the predetermined goal -isshows capital inflow ie flt 0 when Xπ(t)ltL

π(t) and

Xπ(t)ltMπ(t) -erefore the function f(middot middot middot) considersthe average and absolute performance of the wealth processin [t minus h t]

Mathematical Problems in Engineering 3

Substituting (10) into (9) the following stochastic delaydifferential equation (SDDE) is obtained

dπX(t) r minus c1(t) minus c2( 1113857X(t) + c1(t)Lπ(t) + c2Mπ(t) + αp1(t) + c minus δ q1(t) q2(t)( 1113857( 11138571113858 1113859dt

+σp1(t)dW(t) minus q1(t)dC1(t) minus q2(t)dC2(t)

dLπ(t) Xπ(t) minus ALπ(t) minus eminus AhMπ(t)1113960 1113961dt

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)

Furthermore suppose Xπ(t) x0 forallt isin [minus h 0] whichcan be interpreted as that the insurance company has theinitial wealth of x0 at minus h -ere is no business operationduring [minus h 0] and the wealth has no change -e integrateddelay wealth initial value can be calculated to getLπ(0) x0A(1 minus eminus Ah)

Definition 1 (admissible strategy) For any fixed t isin [0 T]an investment-reinsurance strategy π(t) (p1(t) q1(t)

q2(t)) is said to admissible if (i) (p1(t) q1(t) q2(t)) is Ft

progressively measurable (ii) for t isin [0 T] q1(t) isin [0 1]q2(t) isin [0 1] and E[1113938

T

0 p21(t)dt] lt 0 and (iii) SDDE (11) has

a unique strong solution X(middot) such that E[sup0letleT |X(t)|2]

ltinfin Let Π be the set of all admissible investment-rein-surance strategy

3 Optimization Problem

To take historical operating performance into account theinsurer will focus on both terminal wealth Xπ(T) andhistorical average operating performance Lπ(T) thus thefollowing objective function is defined

J(t x l m 1113954π) supπisinΠ

Etxlm Xπ(T) + βL

π(T)1113960 1113961

minusω2Vartxlm X

π(T) + βL

π(T)1113960 1113961

(12)

where risk aversion coefficient ω(gt 0) and delay parameterβ(isin [0 1]) are constants Etxlm[middot] and Vartxlm[middot] representconditional expectation and conditional variance based onXπ(t) x Lπ(t) l andMπ(t) m respectively β(isin [0 1])

is the weight of Lπ(t) indicating the degree of terminal wealth

affected by historical average performance If we writeβ β1113938

0minus h

eAudu then Xπ(t) + βLπ(t) Xπ(t)+ βLπ(t) In

addition according to Chang [18] delay optimal controlproblem is generally an infinite-dimensional problem In orderto obtain the optimal solution some additional conditions willbe attached We assume that the value function V(middot) is onlyrelated tox and l butLπ(t) is related toMπ(t) in order tomakeV(middot) only depend on (t x l) the problem can obtain theoptimal solution and we assume the following conditions hold

c2 βeminus Ah

c1 minus Aβ r minus c1 minus c2 + β( 1113857β(13)

-erefore this paper aims at the following optimizationproblems

J(t x l 1113954π) supπisinΠ

Etxlm Xπ(T) + βL

π(T)1113858 1113859

minusω2Vartxlm X

π(T) + βL

π(T)1113858 1113859

Etxlm F Xπ(T) + βL

π(T)( 11138571113858 1113859

+ G Etxlm Xπ(T) + βL

π(T)1113858 11138591113872 1113873

(14)

where F(x) x minus ω2x2 and G(x) ω2x2

Remark 1

(i) According to Shen and Zeng [25] condition (13)can be regarded as exogenous technical conditionsthat need to be determined in advance by theinsurance company Firstly the average delaywealth L

π(t) and point by point delay wealth

Mπ(t) are determined by selecting the averageparameter A and delay time h Secondly it selectsthe weight β Finally it calculates the weight ratiosc2 βeminus Ah and c1 (β1113938

0minus h

eAudu(1 + β11139380minus h

eAu

du))(r minus c2 + β + A) of historical performanceXπ(t) minus L

π(t) and Xπ(t) minus Mπ(t) according to the

two assumptions in (13) and adjusts the inflowoutflow of capital accordingly

(ii) Because there is a nonlinear function of the ex-pectation of the terminal value wealth in the varianceterm problem (14) is time inconsistent which leadsto the failure of Bellmanrsquos optimal principle Manyworks of literature deal with the mean-varianceproblem by setting a precommitment so the optimalstrategy obtained are time-inconsistent Howeverfor a rational decision maker time consistency isoften not negligible Rational decision makers hopethat the equilibrium strategy they find is not onlyoptimal at this time but also optimal in the futurewith the evolution of time that is to say the equi-librium strategy is time consistent -erefore forproblem (14) this paper aims to find the equilibriumstrategy


Definition 2 Consider a control law 1113954π(t) t isin [0 T] Choose arbitrarily 1113957π isin Π tgt 0 and εgt 0 and define thecontrol law 1113954πε

πε(u) 1113957π tle ult t + ε

1113954π(u) t + εle uleT1113896 (15)

We call that 1113954π is an equilibrium strategy iflimεdarr0inf(J(t x l 1113954π) minus J(t x l πε)ε)ge 0 for any t and 1113957π Ifthe equilibrium strategy 1113954π exists the equilibrium valuefunction is defined as V(t x l) J(t x l 1113954π)

According to Definition 2 the equilibrium strategy istime consistent For simplicity we denote that C121[0 T] times

R times R ϕ(t x l) |ϕ(t middot middot)1113864 1113865 is once continuously differen-tiable on [0 T] ϕ(middot x middot) is twice continuously differentiableon R and ϕ(middot middot l) is once continuously differentiable on RTo provide verification theorem and derive convenientlyextended HJB equation for forall(t x l) isin [0 T] times R times Rforallϕ isin C121([0 T] times R times R) and given control law π wedefine variational operator as follows

Lπϕ(t x l) ϕt(t x l) + r minus c1 minus c2( 1113857x + c1l + c2m + αp1(t) + c minus δ q1(t) q2(t)( 1113857( 11138571113858 1113859ϕx(t x l)

+ x minus Al minus eminus Ah

m1113872 1113873ϕl(t x l) +12σ2p2

1(t)ϕxx(t x l)

+ λ1E ϕ t x minus q1(t)Yi l( 1113857 minus ϕ(t x l)1113858 1113859 + λ2E ϕ t x minus q2(t)Zi l( 1113857 minus ϕ(t x l)1113858 1113859

+ λE ϕ t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus ϕ(t x l)1113858 1113859

(16)

-e following theorem provides verification for theextended HJB equation in problem (14)

Theorem 1 (verification theorem) For problem (14) weassume that there exist two real-valued functionsV(t x l) g(t x l) isin C121([0 T] times R times R) satisfying thefollowing extended HJB equation

supπisinΠ LπV(t x l) minusω2L

πg2(t x l) + ωg(t x l)L

πg(t x l)1113882 1113883 0

L1113954πg(t x l) 0

1113954π argsupπisinΠ LπV(t x l) minusω2L


πg(t x l)1113882 1113883

V(T x l) x + βl g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

-en J(t x l 1113954π) V(t x l) Etxl[X1113954π(T) + βL1113954π(T)]

g(t x l) and 1113954π is an equilibrium investment-reinsurancestrategy

-e proof process of-eorem 1 is similar to that of Bjorket al [28] so it is omitted here

In Definition 1 the policy set Π is allowed to require thereinsurance policy to satisfy the constraint q1(t) isin [0 1] andq2(t) isin [0 1] To facilitate the solution we do not considerthis constraint temporarily and record all the policy setssatisfying (i) and (iii) as 1113954Π According to the variational


operator (16) the extended HJB (17) can be expanded asfollows

supπisin1113954Π

Vt(t x l) + r minus c1(t) minus c2( 1113857x + c1(t)l + c2m + αp1(t) + c minus δ q1(t) q2(t)( 1113857( 11138571113858 1113859Vx(t x l)⎧⎨

⎩

+12σ2p2

1(t) Vxx(t x l) minus ωg2x(t x l)1113960 1113961 + x minus Al minus e

minus Ahm1113872 1113873Vl(t x l) + λ1 E V t x minus q1(t)Yi l( 1113857 minus V(t x l)1113858 11138591113876

minusω2

E g2

t x minus q1(t)Yi l( 1113857 minus g2(t x l)1113960 1113961 + ωg(t x l)E g t x minus q1(t)Yi l( 1113857 minus g(t x l)1113858 11138591113877

+λ2 E V t x minus q2(t)Zi l( 1113857 minus V(t x l)1113858 1113859 minusω2

E g2

t x minus q2(t)Zi l( 1113857 minus g2(t x l)1113960 11139611113876

+ωg(t x l)E g t x minus q2(t)Zi l( 1113857 minus g(t x l)1113858 11138591113877

+λ E V t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus V(t x l)1113858 1113859 minusω2

E g2

t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus g2(t x l)1113960 11139611113876

minus ωg(t x l)E g t x minus x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus g(t x l)1113858 11138591113877⎫⎬

⎭ 0 t isin [0 T]

gt(t x l) + r minus c1(t) minus c2( 1113857x + c1(t)l + c2m + α1113954p1(t) + c minus δ 1113954q1(t) 1113954q2(t)( 1113857( 11138571113858 1113859

gx(t x l) +12σ21113954p

21(t)gxx(t x l) + x minus Al minus e

minus Ahm1113872 1113873gl(t x l) + λ1E g t x minus 1113954q1(t)Yi l( 1113857 minus g(t x l)1113858 1113859

+ λ2E g t x minus 1113954q2(t)Zi l( 1113857 minus g(t x l)1113858 1113859 + λE g t x minus 1113954q1(t)Yi minus 1113954q2(t)Zi l( 1113857 minus g(t x l)1113858 1113859 0 t isin [0 T]

V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

Suppose that the solution of the above extended HJBequation has the following structure

V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)

with the boundary condition H1(T) H2(T) 1 andF1(T) F2(T) 0

Differentiating V and g with respect to t x and l weobtain

Vt(t x l) H1prime(t)(x + βl) + F1prime(t)

Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0

gt(t x l) H2prime(t)(x + βl) + F1prime(t)

gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)

-rough simple calculation we can also obtain

E V t x minus q1(t)Yi l( 1113857 minus V(t x l)1113858 1113859 minus μ1Yq1(t)H1(t)

E V t x minus q2(t)Zi l( 1113857 minus V(t x l)1113858 1113859 minus μ1Zq2(t)H1(t)

E V t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus V(t x l)1113858 1113859 minus μ1Yq1(t)H1(t) minus μ1Zq2(t)H1(t)

⎧⎪⎪⎨

⎪⎪⎩

E g2 t x minus q1(t)Yi l( 1113857 minus g2(t x l)1113858 1113859 μ2Yq21(t) minus 2μ1Yq1(t)(x + βl)1113858 1113859H22(t) minus 2μ1Yq1(t)H2(t)F2(t)

E g2 t x minus q2(t)Zi l( 1113857 minus g2(t x l)1113858 1113859 μ2Zq22(t) minus 2μ1Zq2(t)(x + βl)1113858 1113859H22(t) minus 2μ1Zq1(t)H2(t)F2(t)

E g2 t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus g2(t x l)1113858 1113859 μ2Yq21(t) + μ2Zq22(t) + 2μ1Yμ1Zq1(t)q2(t) minus 2 μ1Yq1(t) + μ1Zq2(t)( 1113857(x + βl)1113858 1113859

H22 minus 2 μ1Yq1(t) + μ1Zq2(t)( 1113857H2(t)F2(t)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

E g t x minus q1(t)Yi l( 1113857 minus g(t x l)1113858 1113859 minus μ1Yq1(t)H2(t)

E g t x minus q2(t)Zi l( 1113857 minus g(t x l)1113858 1113859 minus μ1Zq2(t)H2(t)

E g t x minus q1(t)Yi minus q2(t)Zi l( 1113857 minus g(t x l)1113858 1113859 minus μ1Yq1(t)H2(t) minus μ1Zq2(t)H2(t)

⎧⎪⎪⎨

⎪⎪⎩

(21)


Putting the above results back into (18) we can arrive at

supπisin1113954Π

H1prime(t)(x + βl) + F1prime(t) + ψ p1 q1 q2( 1113857H1(t) minusω2σ2p2

1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t) minus ωλμ1Yμ1Zq1(t)q2(t)H

22(t)1113882 1113883 0

H2prime(t)(x + βl) + F2prime(t) + ψ p1 q1 q2( 1113857H2(t) 0

H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where

ψ p1 q1 q2( 1113857 r minus c1 minus c2 + β( 1113857x + c1 minus Aβ( 1113857l

+ c2 minus βeminus Ah

1113872 1113873m + αp1(t)

+ c minus δ q1(t) q2(t)( 1113857( 1113857 minus a1q1(t) minus a2q2(t)

(23)

According to c2 βeminus Ah we have

ψ p1 q1 q2( 1113857 r minus c1(t) minus c2 + β( 1113857x + c1 minus Aβ( 1113857l + αp1(t)


(24)

For the convenience of writing let

h p q1 q2( 1113857 ψ p q1 q2( 1113857H1(t) minusω2σ2p2

1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)

minus ωλμ1Yμ1Zq1(t)q2(t)H22(t)

(25)

4 Optimal Time-Consistent Strategy

-is section assumes that the reinsurance premium rate iscalculated by the expected premium principle ie

δ q1(t) q2(t)( 1113857 1 + η1( 1113857 1 minus q1(t)( 1113857a1 + 1 + η2( 1113857 1 minus q2(t)( 1113857a2

(26)

where η1 and η2 are the reinsurerrsquos safety loading of theinsurance business

Substituting the above formula into (24) we have

ψ p1 q1 q2( 1113857 r minus c1 minus c2 + β( 1113857x + c1 minus Aβ( 1113857l + αp1(t)

+ c minus a1 1 + η1( 1113857

minus a2 1 + η2( 1113857 + a1η1q1(t) + a2η2q2(t)

(27)

To facilitate derivation we rewrite (25) as


1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)

Differentiating h(p q1 q2) with respect to p1 q1 and q2we can derive

zh

zp1 αH1(t) minus ωσ2p1(t)H

22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh

zq1 a1η1H1(t) minus ωb1q1(t)H

22(t) minus ωλμ1Yμ1Zq2(t)H

22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h

zq1zq2 minus ωλμ1Yμ1ZH

22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

From (29) we obtain the following Hessian matrix


z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus ωH22(t)B (30)

where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)

Lemma 1 Be function h(p1 q1 q2) in (28) is concave withrespect to (p1 q1 q2)

Proof In order to prove Lemma 1 we only need to provethat the Hessian matrix is negative definite From (43) weknow H2(t)ne 0 thus H2

2(t)gt 0 According to (30) we onlyneed to prove that matrix B is positive definiteforallC (c1 c2 c3) isin R3 and Cne 0 Let (middot)tr denote the

transposition of a vector or matrix then

C middot B middot Ctr c

21σ

2+ c

22b1 + c

23b2 + 2c2c3λμ1Yμ1Z

c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)

So matrix B is positive definiteFrom (29) we have

αH1(t) minus ωσ2p1(t)H22(t) 0

a1η1H1(t) minus ωb1q1(t)H22(t) minus ωλμ1Yμ1Zq2(t)H2

2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

By solving the above equations we can obtain

1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

where D1 a1η1b2 minus a2η2λμ1Yμ1Zb1b2 minus λ2μ21Yμ21Z and

D2 a2η2b1 minus a1η1λμ1Yμ1Zb1b2 minus λ2μ21Yμ21Z

From Lemma 1 we know that (1113954p1(t) 1113954q1(t) 1113954q2(t)) is thepoint where function h(p1 q1 q2) takes the maximum valuePutting (1113954p1(t) 1113954q1(t) 1113954q2(t)) into (22) we can obtain

H1prime(t)(x + βl) + F1prime(t) + r minus c1 minus c2 + β( 1113857x + c1 minus Aβ( 1113857l1113858 1113859H1(t)

+ c minus a1 1 + η1( 1113857 minus a2 1 + η2( 11138571113858 1113859H1(t) + α1113954p1(t) + a1η11113954q1(t) + a2η21113954q2(t)1113858 1113859H1(t)

minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)


+ α1113954p1(t) + c minus a1 1 + η1( 1113857 minus a2 1 + η2( 1113857 + a1η11113954q1(t) + a2η21113954q2(t)1113858 1113859H2(t) 0(36)

According to

c1 minus Aβ r minus c1 minus c2 + β( 1113857β (37)we have

r minus c1 minus c2 + β( 1113857x + c1 minus Aβ( 1113857l r minus c1 minus c2 + β( 1113857(x + βl)

(38)

By separating variables of (x + βl) we can obtain


H1prime(t) + r minus c1 minus c2 + β( 1113857H1(t) 0

H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)

F1prime(t) + c minus a1 1 + η1( 1113857 minus a2 1 + η2( 11138571113858 1113859H1(t) + α1113954p1(t) + a1η11113954q1(t) + a2η21113954q2(t)1113858 1113859H1(t)

minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)

F2prime(t) + α1113954p1(t) + c minus a1 1 + η1( 1113857 minus a2 1 + η2( 1113857 + a1η11113954q1(t) + a2η21113954q2(t)1113858 1113859H2(t) 0

F1(T) 01113896 (42)

By solving the above equations we have

H1(t) H2(t) e rminus c1minus c2+β( )(Tminus t)

F1(t) c minus a1 1 + η1( 1113857 minus a2 1 + η2( 1113857

r minus c1 minus c2 + βe

rminus c1minus c2+β( )(Tminus t)minus 11113876 1113877 +

1ω

a1η1D1 + a2η2D2 minus12b1D

21 minus

12b2D

22 minus λμ1Yμ1ZD1D2 +

α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)

According to the above discussion the followingproposition can be obtained

Proposition 1 For problem (14) the time-consistent in-vestment-reinsurance strategy in set 1113954Π is as follows

1113954p1(t) α

σ2ωe

minus rminus c1minus c2+β( )(Tminus t)

1113954q1(t) a1η1b2 minus a2η2λμ1Yμ1Z

b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)

-e corresponding equilibrium function is

V(t x l) H1(t)(x + βl) + F1(t) (45)

where H and F are given by (43)Let t1 T minus (1r minus c1 minus c2 + β)ln(D1ω) for ωleD1 le

ωe(rminus c1minus c2+β)T Let t2 T minus (1r minus c1 minus c2 + β)ln (D2ω) forωleD2 leωe(rminus c1minus c2+β)T For D1 ltω (D2 ltω) we set t1 T

(t2 T) And for D1 gtωe(rminus c1minus c2+β)T (D2 gtωe(rminus c1minus c2+β)T)we set t1 0 (t2 0) To make sure that the optimal

reinsurance strategies satisfy q1(t) isin [0 1] andq2(t) isin [0 1] we introduce the following lemma

Lemma 2 For λ μ1Y μ1Z a1 a2 b1 and b2 given in (4) thefollowing inequality holds

λμ1Yμ1Za2

a1b2leλμ1Yμ1Za2 + b1a2

a1b2 + λμ1Yμ1Za1le

b1a2

λμ1Yμ1Za1 (46)

Proof Using Cauchy minus Schwarz inequality we can easily getb1 gt λμ1Yμ1Z and b2 gt λμ1Yμ1Z and then we can obtain

λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)

In addition for any positive number d1 d2 d3 and d4 if(d1d2)le (d3d4) then (d1d2)le(d1 +d3d2 +d4)le (d3d4)In combination with inequality (47) inequality (46) is easilyproved

From Lemma 2 we will investigate the optimal results inthe following four cases

Case 1 η1 lt (λμ1Yμ1Za2b2a1)η2Case 2 (λμ1Yμ1Za2b2a1)η2 le η1 lt (λμ1Yμ1Za2 + b1a2a1b2 + λμ1Yμ1Za1)η2Case 3 (λμ1Yμ1Za2 + b1a2a1b2 + λμ1Yμ1Za1)η2 le η1 le(b1a2λμ1Yμ1Za1)η2


Case 4 η1 gt (b1a2λμ1Yμ1Za1)

Next the optimal time-consistent strategyπlowast(t) (plowast1 (t) qlowast1 (t) qlowast2 (t)) in admissible strategy set Πand the corresponding value function V(t x l) are dis-cussed In order to have a clear classification discussion it isassumed that r minus c1 minus c2 + βge 0

Case 1 in this case we have 1113954q1(t)lt 0 and 1113954q1(t)ge 0thus qlowast1 (t) 0 Let h1(p1 q2) h(p1 0 q2) Bysubstituting qlowast1 (t) 0 into (28) and maximizingfunction h1(p1 q2) we can get the maximum point

1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2

eminus rminus c1minus c2+β( )(Tminus t)

(48)

Let t3 T minus (1r minus c1 minus c2 + β)ln(η2a2ωb2) For0le tle t3 it is easy to see 1113954q2(t)le 1 and then we haveπlowast(t) (1113954p1(t) 0 q2(t)) Putting (1113954p1(t) 0 q2(t)) into(41) and (45) we obtain

V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where

Q1(t x l) erminus c1minus c2+β( )(Tminus t)

(x + βl)

+c minus a1 1 + η1( 1113857 minus a2 1 + η2( 1113857

r minus c1 minus c2 + β

times erminus c1minus c2+β( )(Tminus t)

minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22

2ωb2(T minus t) (51)

where R1 is a constant whose value will be determinedin a later calculationFor t3 lt tleT we have πlowast(t) (1113954p1(t) 0 1)Substituting it into (41) and (45) we can obtain

V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2


rminus c1minus c2+β( )(Tminus t)minus 11113874 1113875 (53)

Q4(t) minusωb2

4 r minus c1 minus c2 + β( 1113857e2 rminus c1minus c2+β( )(Tminus t)

minus 11113874 1113875

(54)

To make the value function V(t x l) continuous letQ2(t3) + R1 Q3(t3) + Q4(t3) then

R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)

Case 2 in this case we have 1113954q1(t)ge 0 1113954q2(t)ge 0 andD1 leD2 and it is easy to see t2 le t1For 0le tle t2 we have 1113954q1(t)le 1 1113954q2(t)le 1 and thusπlowast(t) (1113954p1(t) 1113954q1(t) 1113954q2(t)) Substituting it into (41)and (45) we can derive

V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874

minus λμ1Yμ1ZD1D21113875(T minus t)

(57)

For tge t2 we have 1113954q2(t)ge 1 and thus qlowast2 (t) 1 Leth2(p1 q1) h(p1 q1 1) Putting qlowast2 (t) 1 into (28)and maximizing function h2(p1 q2) we can get themaximum point

1113954p1(t) α

ωσ2e


1113957q1(t) a1η1eminus rminus c1minus c2+β( )(Tminus t) minus ωλμ1Yμ1Z

ωb1

(58)

Let t4 T minus (1r minus c1 minus c2 + β)ln(a1η1ωλμ1Yμ1Y) andt5 T minus (1r minus c1 minus c2 + β)ln(a1η1ω(b1 + λμ1Yμ1Z))It is easy to see that t4 le t2 le t5For t2 lt tle t5 we have πlowast(t) (1113954p1(t) 1113957q1(t) 1)Inserting it into (41) and (45) we can obtain

V(t x l) Q1(t x l) + Q3(t) + Q6(t) + Q7(t) + R3

(59)

where

Q6(t)a21η21

2ωb1(T minus t) minus

λa1η1μ1Yμ1Z

b1 r minus c1 minus c2 + β( 1113857e

rminus c1minus c2+β( )(Tminus t)minus 11113874 1113875

(60)

Q7(t)ωλ2μ21Yμ21Z

4b1 r minus c1 minus c2 + β( 1113857minus

ωb24 r minus c1 minus c2 + β( 1113857

1113888 1113889

middot e2 rminus c1minus c2+β( )(Tminus t)

1113874 1113875

(61)

For t5 lt tleT we have 1113954q2(t)gt 1 and thusπlowast(t) (1113954p1(t) 1 1) Putting it into (41) and (45) wecan arrive at


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1



Q9(t) minusω b1 + b2 + 2λμ1Yμ1Z( 1113857


minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then

R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857 minus Q3 t5( 1113857 minus Q6 t5( 1113857 minus Q7 t5( 1113857

(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857 minus Q3 t5( 1113857 minus Q6 t5( 1113857 minus Q7 t5( 1113857 minus Q5 t5( 1113857

(67)

Case 3 in this case we have 1113954q1(t)ge 0 1113954q2(t)ge 0 AndD1 geD2 so t1 le t2For 0le tle t1 we have πlowast(t) (1113954p1(t) 1113954q1(t) 1113954q2(t))Substituting it into (41) and (45) we can obtain

V(t x l) Q1(t x l) + Q5(t) + R4 (68)

For tge t1 we have 1113954q1(t)ge 1 and thus qlowast1 (t) 1 De-note by h3(p1 q2) the function h(p1 q1 q2) in (28) Bymaximizing h3(p1 q2) we derive

1113954p1(t) α

ωσ2e



ωb2

(69)

Let t6 T minus (1r minus c1 minus c2 + β)ln(a2η2ωλμ1Yμ1Z) andt7 T minus (1r minus c1 minus c2 + β)ln(a2η2ω(b2+ ωλμ1Yμ1Z))It is easy to see that t6 le t1 le t7For t1 lt tle t7 we have πlowast(t) (1113954p1(t) 1 1113957q2(t)) Bysubstituting it into (41) and (45) we can obtain


(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z

b2 r minus c1 minus c2 + β( 1113857

middot erminus c1minus c2+β( )(Tminus t)

minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)

For t7 lt tleT we have πlowast(t) (1113954p1(t) 1 1) Putting itinto (41) and (45) we can obtain

V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857

+ Q9 t7( 1113857 minus Q8 t7( 1113857 minus Q10 t7( 1113857 minus Q11 t7( 1113857 minus Q5 t1( 1113857

(76)

Case 4 in this case we have 1113954q1(t)ge 0 and 1113954q2(t)lt 0 andthus qlowast2 (t) 0 Let h4(p1 q1) h(p1 q1 q2) Bymaximizing h4(p1 q2) we arrive at

1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)

Let t8 T minus (1r minus c1 minus c2 + β)ln(a1η1ωb1)


For 0le tle t8 we have πlowast(t) (1113954p1(t) q1(t) 0) Insertingit into (41) and (45) we can derive

V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)

From the above discussion we can get the followingtheorem

Theorem 2 Assuming r minus c1 minus c2 + βge 0 the optimal time-consistent investment and reinsurance strategies for problem(14) are as follows

(i) If Case 1 holds the optimal investment-reinsurancestrategies for model (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)

and the value function is given by

V(t x l) Q1(t x l) + Q2(t) + R1 0le tle t3

Q1(t x l) + Q3(t) + Q4(t) t3 lt tleT1113896

(85)

where Q1(t x l) Q2(t) Q3(t) Q4(t) and R1 aregiven by (50)ndash(55) respectively

(ii) If Case 2 holds the optimal investment-reinsurancestrategies for model (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a1η1b2 minus a2η2λμ1Yμ1Z

b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe

minus rminus c1minus c2+β( )(Tminus t)⎛⎝

middota2η2b1 minus a1η1λμ1Yμ1Z

b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe

minus rminus c1minus c2+β( )(Tminus t) 0le tle t2

⎞⎠

ασ2ω


a1η1eminus rminus c1minus c2+β( )(Tminus t) minus ωλμ1Yμ1Z

ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2

Q1(t x l) + Q3(t) + Q6(t) + Q7(t) + R3 t2 lt tle t5

Q1(t x l) + Q3(t) + Q8(t) + Q9(t) t5 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(87)

where Q5(t) Q6(t) Q7(t) Q8(t) Q9(t) R3 and R2are given by (57)ndash(67) respectively

(iii) If Case 3 holds the optimal investment-reinsurancestrategies for model (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω


1a2η2eminus rminus c1minus c2+β( )(Tminus t) minus ωλμ1Yμ1Z

ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)

where Q10(t) Q11(t) R5 and R4 are given by(71)ndash(76) respectively

(iv) If Case 4 holds the optimal investment-reinsurancestrategies for model (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8

Q1(t x l) + Q8(t) + Q13(t) t8 lt tleT

⎧⎪⎨

⎪⎩(91)

where Q12(t) Q13(t) and R6 are given by (79)ndash(83)respectively

Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)

V(t x l) is a continuous function for any(t x l) isin [0 T] times R times R Furthermore


Q2prime t3( 1113857 Q3prime t3( 1113857 + Q4prime t3( 1113857

Q5prime t2( 1113857 Q3prime t2( 1113857 + Q6prime t2( 1113857 + Q7prime t2( 1113857

Q3prime t5( 1113857 + Q6prime t5( 1113857 + Q7prime t5( 1113857 Q3prime t5( 1113857 + Q8prime t5( 1113857 + Q9prime t5( 1113857



Q12prime(t) Q8prime(t) + Q13prime(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)

which includes that V(t x l) is a classical solutionto the extended HJB (18)

(ii) According to -eorem 2 the investment and re-insurance strategy of the insurer is not directlyaffected by the average parameter A and the delaytime h but according to (13) the average parameterA and the delay time h have an indirect influence on

the investment and reinsurance strategy of insur-ance companies

(iii) Note that in the classification discussion of -eo-rem 2 in order to make the classification clear weassume that r minus c1 minus c2 + βge 0 Forr minus c1 minus c2 + βlt 0 we can also make a similardiscussion

When A h β c1 c2 0 problem (14) degener-ates to the case without time delay

Corollary 1 Without time delay the optimal time-consistentinvestment and reinsurance policies of problem (14) are asfollows

(i) If Case 1 holds the optimal investment-reinsurancestrategies for problem (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)


V(t x l) 1113957Q1(t x l) + 1113957Q2(t) + R1 0le tle t3

1113957Q1(t x l) + 1113957Q3(t) + 1113957Q4(t) t3 lt tleT

⎧⎨

⎩

(95)

where

1113957Q1(t x l) er(Tminus t)(x + βl) +c minus a1 1 + η1( 1113857 minus a2 1 + η2( 1113857

re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873

1113957Q4(t) minusωb2

4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)


(ii) If Case 2 holds the optimal investment-reinsurancestrategies for problem (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe

minus r(Tminus t)a2η2b1 minus a1η1λμ1Yμ1Z

b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe

minus r(Tminus t)⎛⎝ ⎞⎠ 0le tle t2

ασ2ω

eminus r(Tminus t)

a1η1eminus r(Tminus t) minus ωλμ1Yμ1Z

ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2

1113957Q1(t x l) + 1113957Q3(t) + 1113957Q6(t) + 1113957Q7(t) + R3 t2 lt tle t5

1113957Q1(t x l) + 1113957Q3(t) + 1113957Q8(t) + 1113957Q9(t) t5 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D

22 minus λμ1Yμ1ZD1D21113874 1113875(T minus t)

1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873

1113957Q9(t) minusω b1 + b2 + 2λμ1Yμ1Z( 1113857

4re2r(Tminus t)

minus 11113872 1113873

1113957R3 1113957Q3 t5( 1113857 + 1113957Q8 t5( 1113857 + 1113957Q9 t5( 1113857 minus 1113957Q3 t5( 1113857 minus 1113957Q6 t5( 1113857 minus 1113957Q7 t5( 1113857

1113957R2 1113957Q3 t2( 1113857 + 1113957Q6 t2( 1113857 + 1113957Q7 t2( 1113857 + 1113957Q3 t5( 1113857 + 1113957Q8 t5( 1113857 + 1113957Q9 t5( 1113857 minus 1113957Q3 t5( 1113857 minus 1113957Q6 t5( 1113857 minus 1113957Q7 t5( 1113857 minus 1113957Q5 t2( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)


(iii) If Case 3 holds the optimal investment-reinsurancestrategies for problem (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)

1a2η2eminus r(Tminus t) minus ωλμ1Yμ1Z

ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)

(iv) If Case 4 holds the optimal investment-reinsurancestrategies for problem (14) are

plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)

5 Numerical Simulations

In this section Example 1 will be used to illustrate thespecific numerical calculation process of finding the optimal


time-consistent strategy and Example 2 will be used toanalyze the influence of important parameters on the op-timal time-consistent strategy Assuming that the claimamount Yi and Zi are exponentially distributed with pa-rameters ξ1 and ξ2 respectively then μ1Y 1ξ1 μ1Z 1ξ2b1 2(λ + λ1)ξ

21 and b2 2(λ + λ2)ξ

22

Example 1 Let η1 η2 07 ξ1 2 ξ2 3 λ 2 λ1 3λ2 5 α1 05 σ 02 T 8 r 018 β 01 A 015and h 02 and according to Remark 1 we can calculatec1 00064 and c2 00970 and thus r minus c1 minus c2 + β

01765gt 0 According to the above model parameters Ta-ble 1 can be calculated

From Table 1 for tge 5 we have 1113954q2(t)gt 1 According tothe analysis of -eorem 2 it is easy to see that t2 T minus

(1r minus c1 minus c2 + β)ln(D2ω) 28762 and t5 T minus (1rminus

c1 minus c2 + β)ln(a1η1 ω(b1 + λμ1Yμ1Z)) 68029 t2 lt tle t5and hence qlowast1 (t) a1η1eminus (rminus c1minus c2+β)(Tminus t) minus ωλμ1Yμ1Zωb1For t5 lt tleT we have qlowast1 (t) 1 So recalculate Table 1 toobtain Table 2

Example 2 If there is no special description in this examplethe basic parameter values are as follows η1 η2 07ξ1 2 ξ2 3 λ 3 λ1 2 λ2 4 α1 05 σ 02r 018 A 01 β 01 h 02 and ω 05

Figures 1 and 2 depict the influence of risk aversionparameter ω and delay parameter β on the optimal time-consistent investment strategy From Figure 1 we can seethat the optimal time-consistent investment strategy p1(t)

decreases with the increase of risk aversion parameter ω thatis to say the higher the risk aversion degree of the insurer isthe less the amount of risk investment will be Becauseparameter β includes the information of average parameterA and delay h it is a comprehensive time-delay parameterso we only analyze β Figure 2 shows that the larger the delayparameter β is the larger the number of investment in riskyassets will be Note that if β 0 then the insurer decision-making is only based on the current information so it maytake short-term risk-taking behavior for the immediatepossible high return For βgt 0 when the insurer is makingdecision the comprehensive performance in the past periodwill be taken into account Insurer focuses on information ina period when making decisions According to (12) thegreater the value of β the greater the proportion of average

Table 2 Optimal time-consistent strategy in Π

t 0 1 2 3 4 5 6 7 8plowast1 19205 25032 32628 42528 55433 72253 94177 122753 160000qlowast1 04213 05026 05997 04458 05577 06911 08502 10000 10000qlowast2 06019 07181 08567 10000 10000 10000 10000 10000 10000

0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1

Figure 1 -e effect of risk aversion parameter ω on p1

0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03

Figure 2 -e effect of delay parameter β on p1

Table 1 Optimal time-consistent strategy in 1113954Π

t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706


wealth in performance measurement-at is the insurer canchange the inflowoutflow of the insurerrsquos capital byadjusting the size of the parameter beta thus changing therisk faced by the insurer -e bigger the beta the smaller therisk so the insurer will consider increasing the number ofrisky assets

Figures 3ndash6 depict the influence of risk aversion coef-ficient ω and delay parameter β on two types of insurancereinsurance According to Figures 3 and 4 q1(t) and q2(t)

decrease with respect to ω -e higher the risk aversiondegree of the insurer the more reinsurance he will buy toreduce his risk so the retention ratio of q1(t) and q2(t) willbe reduced Figures 5 and 6 show that the retention ratio

q1(t) (q1(t)) increase with respect to the parameter β As theimpact of β on investment strategy p1 -e larger the β thestronger the insurerrsquos ability to adjust capital inflowoutflowthat is the stronger the insurerrsquos risk control ability To acertain extent the profitability of the insurer will be strongerso the insurer will reduce the purchase of reinsurance andthe proportion of reinsurance retention q1(t) (q1(t)) willincrease -is is consistent with economic reality which themore information investors observe the more profit theywill make

Figures 7ndash9 depict the effect of the claim intensity λ1 λ2and λ on reinsurance In Figure 7 the larger the λ1 is thelarger the q1(t) is and the smaller the q2(t) is Because the

02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1

Figure 4 -e effect of risk aversion parameter ω on q2

t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03

Figure 5 -e effect of delay parameter β on q1

q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03



larger the λ1 is the greater the expected claim amount of thefirst type of insurance business will be so the insurer willpurchase more reinsurance for the first type of insurancebusiness and reduce the proportion of retained insuranceq1(t) At this time λ2 will remain unchanged that is theexpected claim amount of the second type of insurancebusiness will remain unchanged Based on the considerationof constant total risk and more profits the insurer willincrease the retention ratio q2(t) of reinsurance A similaranalysis can explain why with the increase of λ2 q1(t)

decreases and q2(t) increases in Figures 8 and 9 which showsthat the retention ratios q1(t) and q2(t) of the two types ofinsurance businesses decrease with the increase of lambdaBecause the larger the lambda is the greater the expectedclaim amount of the two types of insurance businesses willbe -erefore in order to control the risk within a certain

range the insurer will buy more reinsurance for the twotypes of insurance businesses and reduce the retention ratioq1(t) and q2(t)

6 Conclusion

In this paper we study the optimal investment-reinsuranceproblem with delay and risk dependence under the mean-variance preference criterion Considering the time-delay effectand risk dependence we obtain the extendedHJB equation withdelay based on the time delay stochastic control framework andthe equilibrium stochastic controlmethod-e results show thatthe optimal time-consistent investment and reinsurance strategywill be affected by the time delay effect -e larger the capitalflow related to the historical business performance the greaterthe risk faced by the insurance company In a prudent attitudethe insurer will reduce the amount invested in a risk asset andreduce the reinsurance retention ratio of all insurance busi-nesses In addition risk dependence is linked by common riskshock sources -e greater the risk common shock intensity isthe smaller the reinsurance retention ratio will be From thenumerical analysis results we can see not only the numericalcalculation process of the optimal strategy but also the intuitiveverification of the above conclusions

In this paper we study the risk assets under geometricBrownian motion To better simulate the real financialmarket the following research will consider the introductionof CEV Heston and other stochastic volatility modelsVasicek CIR and other stochastic interest rate models

Data Availability

-e data in this paper can be used publicly

Conflicts of Interest

-e authors declare that they have no conflicts of interest

02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2

Figure 7 -e effect of λ1 on q

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2

Figure 9 -e effect of λ on q


Acknowledgments

-is work was supported by the science and technologyresearch project of Chongqing Education Commissionunder Grant KJQN201801529 and Doctoral ResearchProjects for Central Universities under Grant JBK2007190

References

[1] S Browne ldquoOptimal investment policies for a firm with arandom risk process exponential utility and minimizing theprobability of ruinrdquo Mathematics of Operations Researchvol 20 no 4 pp 937ndash958 1995

[2] C Hipp and M Plum ldquoOptimal investment for insurersrdquoInsurance Mathematics and Economics vol 27 no 2pp 215ndash228 2000

[3] Z Wang J Xia and L Zhang ldquoOptimal investment for aninsurer the martingale approachrdquo Insurance Mathematicsand Economics vol 40 no 2 pp 322ndash334 2007

[4] C S Liu and H Yang ldquoOptimal investment for an insurer tominimize its probability of ruinrdquo North American ActuarialJournal vol 8 no 2 pp 11ndash31 2004

[5] H Yang and L Zhang ldquoOptimal investment for insurer withjump-diffusion risk processrdquo Insurance Mathematics andEconomics vol 37 no 3 pp 615ndash634 2005

[6] L Bai and J Guo ldquoOptimal proportional reinsurance andinvestment with multiple risky assets and no-shorting con-straintrdquo Insurance Mathematics and Economics vol 42 no 3pp 968ndash975 2008

[7] S David Promislow and V R Young ldquoMinimizing theprobability of ruin when claims follow brownian motion withdriftrdquo North American Actuarial Journal vol 9 no 3pp 110ndash128 2005

[8] N Bauerle ldquoBenchmark and mean-variance problems forinsurersrdquo Mathematical Methods of Operations Researchvol 62 no 1 pp 159ndash165 2005

[9] Y Zeng and Z Li ldquoOptimal time-consistent investment andreinsurance policies for mean-variance insurersrdquo InsuranceMathematics and Economics vol 49 no 1 pp 145ndash154 2011

[10] Z Liang K C Yuen and J Guo ldquoOptimal proportionalreinsurance and investment in a stock market with Ornstein-Uhlenbeck processrdquo Insurance Mathematics and Economicsvol 49 no 2 pp 207ndash215 2011

[11] Z Liang K C Yuen and J Guo ldquoOptimal control of excess-of-loss reinsurance and investment for insurers under a cevmodelrdquo Insurance Mathematics and Economics vol 51p 674 2012

[12] K C Yuen Z Liang and M Zhou ldquoOptimal proportionalreinsurance with common shock dependencerdquo InsuranceMathematics and Economics vol 64 pp 1ndash13 2015

[13] J Grandell Aspects of Risk Beory Springer-Verlag NewYork NY USA 1991

[14] Z Liang and K C Yuen ldquoOptimal dynamic reinsurance withdependent risks variance premium principlerdquo ScandinavianActuarial Journal vol 2016 no 1 pp 18ndash36 2016

[15] Z Ming Z Liang and C Zhang ldquoOptimal mean-variancereinsurance with common shock dependencerdquo Be AnziamJournal vol 58 no 2 pp 162ndash181 2016

[16] J Bi Z Liang and F Xu ldquoOptimal mean-variance investmentand reinsurance problems for the risk model with commonshock dependencerdquo Insurance Mathematics and Economicsvol 70 pp 245ndash258 2016

[17] J Bi and K Chen ldquoOptimal investment-reinsurance problemswith common shock dependent risks under two kinds of

premium principlesrdquo RAIRO - Operations Research vol 53no 1 pp 179ndash206 2019

[18] M-H Chang T Pang and Y Yang ldquoA stochastic portfoliooptimization model with bounded memoryrdquo Mathematics ofOperations Research vol 36 no 4 pp 604ndash619 2011

[19] S Federico ldquoA stochastic control problem with delay arisingin a pension fund modelrdquo Finance and Stochastics vol 15no 3 pp 421ndash459 2011

[20] X Peng W Su and Z Zhang ldquoOn a perturbed compoundPoisson risk model under a periodic threshold-type dividendstrategyrdquo Journal of Industrial and Management Optimiza-tion vol 13 no 5 pp 1ndash20 2017

[21] W Yu P Guo QWang et al ldquoOn a periodic capital injectionand barrier dividend strategy in the compound Poisson riskmodelrdquo Mathematics vol 8 no 4 p 511 2020

[22] I Elsanosi B Oslashksendal and A Sulem ldquoSome solvable sto-chastic control problems with delayrdquo Stochastics and Sto-chastic Reports vol 71 no 1-2 pp 69ndash89 2000

[23] B Oslashksendal and A Sulem ldquoAmaximum principle for optimalcontrol of stochastic systems with delay with applications tofinancerdquo 2000

[24] D David ldquoOptimal control of stochastic delayed systems withjumpsrdquo 2020

[25] Y Shen and Y Zeng ldquoOptimal investment-reinsurancestrategy for mean-variance insurers with square-root factorprocessrdquo Insurance Mathematics and Economics vol 62pp 118ndash137 2015

[26] Y Lai ldquoOptimal investment and excess-of-loss reinsuranceproblem with delay for an insurer under Hestonrsquos SV modelrdquoInsurance Mathematics and Economics vol 61 pp 181ndash1962015

[27] Y Lai and Y Shao ldquoOptimal excess-of-loss reinsurance andinvestment problem with delay and jump-diffusion riskprocess under the CEVmodelrdquo Journal of Computational andApplied Mathematics vol 342 pp 317ndash336 2018

[28] T Bjork M Khapko and A Murgoci ldquoOn time-inconsistentstochastic control in continuous timerdquo Finance and Sto-chastics vol 21 no 2 pp 331ndash360 2017


return of stocks and obtained the optimal reinsurance andinvestment strategy Gu et al [11] investigate the excess-of-loss reinsurance-investment problem under the constantelasticity variance (CEV) model

-ere are two deficiencies in the above literature thatdeserve further discussion On the one hand these literatureimplicitly assume that all insurance businesses of insurersare independent of each other so they only study the in-vestment and reinsurance of a single insurance businessHowever in the real insurance market there are often in-terdependencies between insurance businesses For exampleduring the 2019-nCoV medical claims and death claimsoften occur together In order to depict this kind of de-pendency between different insurance businesses the riskdependent model is proposed -e main works in this areaare as follows Yuen et al [12] taking the expected utilitymaximization of the terminal wealth as the criterion con-sidered the optimal proportional reinsurance problem withmultidimensional risk dependence by using the diffusionapproach method For the detailed process of diffusionapproximate to the compound Poisson process see Gandell[13] Liang and Yuen [14] under the principle of variancepremium investigated the optimal proportional reinsuranceof the Poisson model and diffusion approximation modelMing et al [15] derive the explicit expression of the optimalproportional reinsurance under the mean-variance criterionby using stochastic linear quadratic control Considering thecombination of investment and reinsurance Bi et al [16]obtains the optimal investment-reinsurance strategy formean-variance under the diffusion approximation model Biand Chen [17] under the criterion of maximizing the ex-pected utility of terminal wealth arrived at the optimalinvestment and reinsurance strategies On the other handmost of the literature on optimal investment-reinsuranceand other optimal control problems focus on time-delay freecontrolled systems In fact financial markets tend to rely onthe past Chang [18] considers the investment and con-sumption problems related to the return on risk assets andthe historical performance Federico [19] introduces thetime-delay state process by considering the capital inflowoutflow related to performance Peng et al [20] and Yu et al[21] study the optimal dividend policy based on observingthe information of past time points to determine the be-havior of the next moment In fact this is a discrete case oftime delay However the stochastic control problems ofsystems with time-delay state may be infinite-dimensional incontinuous cases hence it is difficult to find the analyticalsolutions Only in some special cases it is finite-dimensionaland the problem has explicit solution Elsanosi et al [22]emptyksendal and Sulem [23] and David [24] provide a theo-retical basis for solving such problems Shen and Zeng [25]first introduced the time delay in the investment and in-surance problem -ey introduced the inflowoutflow ofcapital in the wealth process of insurer and then depicted thewealth process of insurance companies through the sto-chastic delay differential equation (SDDE) After that Li [26]and Lai [27] studied the optimal investment-reinsuranceproblem with time delay under Heston and CEV modelsrespectively

Inspired by the above research this paper combines riskdependence with time delay to consider investment-rein-surance problem -e structure of the rest of this paper is asfollows In Section 2 the financial model framework of thispaper is given assuming that an insurer can invest in a risk-free asset and a risky asset and in the case of two-dimen-sional dependent claim compound Poisson model and theintroduction of the historical performance of the insurancecompany the companyrsquos wealth process with time delay isobtained In Section 3 considering the mean-variancepreference criterion the time-inconsistent optimizationproblem is defined and the extended HJB equation is ob-tained by using the stochastic control theory in theframework of game theory In Section 4 under the principleof reinsurance expected premium the explicit solutions ofoptimal investment and reinsurance strategies and theircorresponding value functions are derived In Section 5 thenumerical calculation process of optimal investment andreinsurance strategies are introduced through numericalexamples and the influence of important model parameterson optimal strategy is analyzed Section 6 concludes thispaper

2 The Model

Suppose that model is based on the probability space(ΩF F tisin[0T]P) of information flow which satisfies thegeneral assumptions of right continuity and completenesswhere T is a finite constant representing the operation cycleof an insurance company and F tisin[0T] is the sum of in-formation available up to time t All stochastic processesinvolved in this paper are assumed to adapt to F tisin[0T]

Suppose an insurer has an insurance portfolio businesswhich is composed of two different insurance businesses suchas medical insurance and death insurance Suppose that therandom variable Yi ige 11113864 1113865 represents the claim amount of thefirst type of insurance business they are independent and havethe same distribution function FY(y) Zi ige 11113864 1113865 represents theclaim amount of the second type of insurance business they areindependent and have the same distribution function FZ(z)We assume that if yle 0 then FY(y) 0 Otherwise0ltFY(y)le 1 And also assume that if zle 0 then FZ(z) 0Otherwise 0ltFZ(z)le 1 In addition their moment gener-ating functions MY(ι) and MZ(ι) exist -e cumulative claimprocess of the two insurance businesses are as follows

C1(t) 1113944

1113957N1(t)

i1Yi

C2(t) 1113944

1113957N2(t)

i1Zi

(1)

where 1113957N1(t) and 1113957N2(t) represent the number of claims forthe first and second categories of insurance business up totime t respectively And suppose Yi ige 11113864 1113865 Zi ige 11113864 1113865

1113957N1(t)1113864 1113865tgt 0 and 1113957N2(t)1113864 1113865tgt 0 are independent of each otherFor different insurance businesses it is assumed that

they are interdependent as follows


1113957N1(t) N1(t) + N(t)

1113957N1(t) N2(t) + N(t)(2)


C(t) C1(t) + C2(t) 1113944

N1(t)+N(t)

i1Yi + 1113944

N2(t)+N(t)

i1Zi (3)




a1 ≔ E C1(t)1113858 1113859 λ + λ1( 1113857μ1Y

b1 ≔ Var C21(t)1113960 1113961 λ + λ1( 1113857μ2Y

a2 ≔ E C2(t)1113858 1113859 λ + λ2( 1113857μ1Z

b2 ≔ Var C22(t)1113960 1113961 λ + λ2( 1113857μ2Z

(4)


μ2Z E[Z2i ]



B(0) 11113896 (5)



S(0) s01113896 (6)







dXπ(t) rX



(8)




dXπ(t) rX



π(t) X

π(t) minus M

π(t)1113872 1113873dt


(9)


f t Xπ(t) minus L

π(t) X

π(t) minus M

π(t)1113872 1113873

c1 Xπ(t) minus L

π(t)1113872 1113873 + c2 X

π(t) minus M

π(t)( 1113857

c1 Xπ(t) minus

Lπ(t)

11139460

minus he

As

ds⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + c2 Xπ(t) minus M

π(t)( 1113857

c1 + c2( 1113857Xπ(t) minus c1L

π(t) minus c2M

π(t)

(10)


eAuduLπ(t) 1113938

0minus h


0minus h





π(t) and


π(t) and







⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)





T







Etxlm Xπ(T) + βL

π(T)1113960 1113961

minusω2Vartxlm X

π(T) + βL

π(T)1113960 1113961

(12)




0minus h



c2 βeminus Ah




Etxlm Xπ(T) + βL

π(T)1113858 1113859

minusω2Vartxlm X

π(T) + βL

π(T)1113858 1113859


π(T)( 11138571113858 1113859


π(T)1113858 11138591113872 1113873

(14)


Remark 1




0minus h


eAu








1113954π(u) t + εle uleT1113896 (15)






m1113872 1113873ϕl(t x l) +12σ2p2

1(t)ϕxx(t x l)



(16)





πg(t x l)1113882 1113883 0

L1113954πg(t x l) 0



πg(t x l)1113882 1113883


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)







supπisin1113954Π


⎩

+12σ2p2



minusω2

E g2



E g2




E g2



⎭ 0 t isin [0 T]


gx(t x l) +12σ21113954p




V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)


V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)




Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0


gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)





⎧⎪⎪⎨

⎪⎪⎩





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩




⎧⎪⎪⎨

⎪⎪⎩

(21)



supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References































1113957N1(t) N1(t) + N(t)

1113957N1(t) N2(t) + N(t)(2)


C(t) C1(t) + C2(t) 1113944

N1(t)+N(t)

i1Yi + 1113944

N2(t)+N(t)

i1Zi (3)




a1 ≔ E C1(t)1113858 1113859 λ + λ1( 1113857μ1Y

b1 ≔ Var C21(t)1113960 1113961 λ + λ1( 1113857μ2Y

a2 ≔ E C2(t)1113858 1113859 λ + λ2( 1113857μ1Z

b2 ≔ Var C22(t)1113960 1113961 λ + λ2( 1113857μ2Z

(4)


μ2Z E[Z2i ]



B(0) 11113896 (5)



S(0) s01113896 (6)







dXπ(t) rX



(8)




dXπ(t) rX



π(t) X

π(t) minus M

π(t)1113872 1113873dt


(9)


f t Xπ(t) minus L

π(t) X

π(t) minus M

π(t)1113872 1113873

c1 Xπ(t) minus L

π(t)1113872 1113873 + c2 X

π(t) minus M

π(t)( 1113857

c1 Xπ(t) minus

Lπ(t)

11139460

minus he

As

ds⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ + c2 Xπ(t) minus M

π(t)( 1113857

c1 + c2( 1113857Xπ(t) minus c1L

π(t) minus c2M

π(t)

(10)


eAuduLπ(t) 1113938

0minus h


0minus h





π(t) and


π(t) and







⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)





T







Etxlm Xπ(T) + βL

π(T)1113960 1113961

minusω2Vartxlm X

π(T) + βL

π(T)1113960 1113961

(12)




0minus h



c2 βeminus Ah




Etxlm Xπ(T) + βL

π(T)1113858 1113859

minusω2Vartxlm X

π(T) + βL

π(T)1113858 1113859


π(T)( 11138571113858 1113859


π(T)1113858 11138591113872 1113873

(14)


Remark 1




0minus h


eAu








1113954π(u) t + εle uleT1113896 (15)






m1113872 1113873ϕl(t x l) +12σ2p2

1(t)ϕxx(t x l)



(16)





πg(t x l)1113882 1113883 0

L1113954πg(t x l) 0



πg(t x l)1113882 1113883


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)







supπisin1113954Π


⎩

+12σ2p2



minusω2

E g2



E g2




E g2



⎭ 0 t isin [0 T]


gx(t x l) +12σ21113954p




V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)


V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)




Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0


gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)





⎧⎪⎪⎨

⎪⎪⎩





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩




⎧⎪⎪⎨

⎪⎪⎩

(21)



supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References



































⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)





T







Etxlm Xπ(T) + βL

π(T)1113960 1113961

minusω2Vartxlm X

π(T) + βL

π(T)1113960 1113961

(12)




0minus h



c2 βeminus Ah




Etxlm Xπ(T) + βL

π(T)1113858 1113859

minusω2Vartxlm X

π(T) + βL

π(T)1113858 1113859


π(T)( 11138571113858 1113859


π(T)1113858 11138591113872 1113873

(14)


Remark 1




0minus h


eAu








1113954π(u) t + εle uleT1113896 (15)






m1113872 1113873ϕl(t x l) +12σ2p2

1(t)ϕxx(t x l)



(16)





πg(t x l)1113882 1113883 0

L1113954πg(t x l) 0



πg(t x l)1113882 1113883


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)







supπisin1113954Π


⎩

+12σ2p2



minusω2

E g2



E g2




E g2



⎭ 0 t isin [0 T]


gx(t x l) +12σ21113954p




V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)


V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)




Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0


gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)





⎧⎪⎪⎨

⎪⎪⎩





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩




⎧⎪⎪⎨

⎪⎪⎩

(21)



supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References

































1113954π(u) t + εle uleT1113896 (15)






m1113872 1113873ϕl(t x l) +12σ2p2

1(t)ϕxx(t x l)



(16)





πg(t x l)1113882 1113883 0

L1113954πg(t x l) 0



πg(t x l)1113882 1113883


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)







supπisin1113954Π


⎩

+12σ2p2



minusω2

E g2



E g2




E g2



⎭ 0 t isin [0 T]


gx(t x l) +12σ21113954p




V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)


V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)




Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0


gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)





⎧⎪⎪⎨

⎪⎪⎩





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩




⎧⎪⎪⎨

⎪⎪⎩

(21)



supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































supπisin1113954Π


⎩

+12σ2p2



minusω2

E g2



E g2




E g2



⎭ 0 t isin [0 T]


gx(t x l) +12σ21113954p




V(T x l) x + βl

g(T x l) x + βl

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)


V(t x l) H1(t)(x + βl) + F1(t)

g(t x l) H2(t)(x + βl) + F2(t)1113896 (19)




Vx(t x l) H1(t)

Vl(t x l) βH1(t)

Vxx(t x l) 0


gx(t x l) H2(t)

gl(t x l) βH2(t)

gxx(t x l) 0

(20)





⎧⎪⎪⎨

⎪⎪⎩





⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩




⎧⎪⎪⎨

⎪⎪⎩

(21)



supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































supπisin1113954Π


1(t)H22(t) minus

ω2

b1q21(t) + b2q

22(t)1113872 1113873H


22(t)1113882 1113883 0


H1(T) H2(T) 1

F1(T) F2(T) 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(22)

where



1113872 1113873m + αp1(t)


(23)




(24)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(25)




(26)




+ c minus a1 1 + η1( 1113857


(27)



1(t)H22(t)

minusω2

b1q21(t) + b2q

22(t)1113872 1113873H

22(t)


(28)


zh


22(t)

z2h

zp21

minus ωσ2H22(t)

z2h

zp1zq1

z2h

zp1zq2 0

zh



22(t)

zh



22(t)

z2h

zq21 minus ωb1H

22(t)

z2h

zq22 minus ωb2H

22(t)

z2h


22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)



z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References































z2h

zp21

z2h

zp1zq1

z2h

zp1zq2

z2h

zp1zq1

z2h

zq21

z2h

zq1zq2

z2h

zp1zq2

z2h

zq1zq2

z2h

zq22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


where

B

σ2 0 0

0 b1 λμ1Yμ1Z

0 λμ1Yμ1Z b2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (31)






21σ

2+ c

22b1 + c


c21σ

2+ c

22 λ1 + λ( 1113857μ2Y + c

23 λ2 + λ( 1113857μ2Z + 2c2c3μ1Yμ1Z

c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c

22E Yi( 1113857

21113960 1113961 + c

23E Zi( 1113857

21113960 1113961 + 2c2c3E Yi1113858 1113859E Yi1113858 11138591113960 1113961

ge c1σ2

+ c22λ1E Yi( 1113857

21113960 1113961 + c

23λ2E Zi( 1113857

21113960 1113961 + λ c2E Yi1113858 1113859 + c3E Zi1113858 11138591113858 1113859

2 gt 0

(32)




2(t) 0


2(t) 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)


1113954p1(t) ασ2

H1(t)

ωH22(t)

1113954q1(t) D1H1(t)

ωH22(t)

1113954q2(t) D2H1(t)

ωH22(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)






minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t)1113872 1113873 + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

(35)



According to



(38)




H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































H1(T) 1

⎧⎪⎨

⎪⎩(39)


H2(T) 11113896 (40)


minusω2

σ21113954p21(t) + b11113954q

21(t) + b21113954q

22(t) + 2λμ1Yμ1Z1113954q1(t)1113954q2(t)1113960 1113961H

22(t) 0

F1(T) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(41)


F1(T) 01113896 (42)






1ω


21 minus

12b2D


α2

2σ21113890 1113891(T minus t)




1ω

a1η1D1 + a2η2D2 +α2

σ21113890 1113891(T minus t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(43)



1113954p1(t) α

σ2ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(44)


V(t x l) H1(t)(x + βl) + F1(t) (45)






λμ1Yμ1Za2



b1a2

λμ1Yμ1Za1 (46)


λμ1Yμ1Za2

b2a1le

b1a2

λμ1Yμ1Za1 (47)








1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References


































1113954p1(t) α

ωσ2e


q2(t) a2η2ωb2


(48)


V(t x l) Q1(t x l) + Q2(t) + R1 (49)

where


(x + βl)




minus 11113874 1113875 +α2

2ωσ2(T minus t)

(50)

Q2(t) a22η

22



V(t x l) Q1(t x l) + Q3(t) + Q4(t) (52)

where

Q3(t) a2η2



Q4(t) minusωb2


minus 11113874 1113875

(54)


R1 Q3 t3( 1113857 + Q4 t3( 1113857 minus Q2 t3( 1113857 (55)


V(t x l) Q1(t x l) + Q5(t) + R2 (56)

where

Q5(t) 1ω


21 minus

12b2D

221113874


(57)


1113954p1(t) α

ωσ2e



ωb1

(58)



(59)

where

Q6(t)a21η21


λa1η1μ1Yμ1Z



(60)




1113888 1113889


1113874 1113875

(61)



V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References































V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (62)

where

Q8(t) a1η1





minus 11113874 1113875

(64)

Let

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857

+ Q8 t5( 1113857 + Q9 t5( 1113857

(65)

then


(66)

R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + Q3 t5( 1113857


(67)


V(t x l) Q1(t x l) + Q5(t) + R4 (68)


1113954p1(t) α

ωσ2e



ωb2

(69)



(70)

where

Q10(t) a22η22


λa2η2μ1Yμ1Z



minus 11113874 1113875

(71)

Q11(t) ωλ2μ21Yμ

21Z


ωb1

4 r minus c1 minus c2 + β( 11138571113888 1113889


1113874 1113875

(72)


V(t x l) Q1(t x l) + Q3(t) + Q8(t) + Q9(t) (73)

Let

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5

Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

(74)

We derive


(75)

R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + Q3 t7( 1113857 + Q8 t7( 1113857


(76)


1113954p1(t) α

ωσ2e


q1(t) a2η2ωb2


(77)




V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































V(t x l) Q1(t x l) + Q12(t) + R6 (78)

where

Q12(t) a21η

21



V(t x l) Q1(t x l) + Q8(t) + Q13(t) (80)

where

Q13(t) minusωb1


minus 11113874 1113875

(81)

Let

Q12(t) + R6 Q8(t) + Q13(t) (82)

We have

R6 Q8(t) + Q13(t) minus Q12(t) (83)




plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2


0a2η2ωb2


1113888 1113889 0le tle t3

αωσ2


0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(84)




(85)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb1 11113888 1113889 t2 lt tle t5

ασ2ω


1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(86)



V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References































V(t x l)

Q1(t x l) + Q5(t) + R2 0le tle t2



⎧⎪⎪⎨

⎪⎪⎩

(87)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe



b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


⎞⎠

ασ2ω



ωb21113888 1113889 t1 lt tle t7

ασ2ω


1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(88)


V(t x l)

Q1(t x l) + Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎨

⎪⎪⎩

(89)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω


a2η2ωb2


01113888 1113889 0le tle t8

ασ2ω


1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(90)


Q1(t x l) + Q12(t) + R6 0le tle t8


⎧⎪⎨

⎪⎩(91)


Remark 2 (i) Since

Q2 t3( 1113857 + R1 Q3 t3( 1113857 + Q4 t3( 1113857

Q5 t2( 1113857 + R2 Q3 t2( 1113857 + Q6 t2( 1113857 + Q7 t2( 1113857 + R3

Q3 t5( 1113857 + Q6 t5( 1113857 + Q7 t5( 1113857 + R3 Q3 t5( 1113857 + Q8 t5( 1113857 + Q9 t5( 1113857

Q5 t1( 1113857 + R4 Q8 t1( 1113857 + Q10 t1( 1113857 + Q11 t1( 1113857 + R5

Q8 t7( 1113857 + Q10 t7( 1113857 + Q11 t7( 1113857 + R5 Q3 t7( 1113857 + Q8 t7( 1113857 + Q9 t7( 1113857

Q12(t) + R6 Q8(t) + Q13(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(92)









⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References





































⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(93)








plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

αωσ2

eminus r(Tminus t)

0a2η2ωb2

eminus r(Tminus t)

1113888 1113889 0le tle t3

αωσ2

eminus r(Tminus t)

0 11113874 1113875 t3 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(94)




⎧⎨

⎩

(95)

where


re

r(Tminus t)minus 11113872 1113873 +

α2

2ωσ2(T minus t)

1113957Q2(t) a22η

22

2ωb2(T minus t)

1113957Q3(t) a2η2

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873

1113957R1 1113957Q3 t3( 1113857 + 1113957Q4 t3( 1113857 minus 1113957Q2 t3( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(96)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb1 11113888 1113889 t2 lt tle t5

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t5 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(97)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R2 0le tle t2



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(98)

where

1113957Q5(t) 1ω


21 minus

12b2D


1113957Q6(t) a21η

21


λa1η1μ1Yμ1Z

b1re

r(Tminus t)minus 11113872 1113873

1113957Q7(t) ωλ2μ21Yμ21Z

4b1rminusωb2

4r1113888 1113889 e2r(Tminus t)( 1113857

1113957Q8(t) a1η1

re

r(Tminus t)minus 11113872 1113873


4re2r(Tminus t)

minus 11113872 1113873



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(99)



plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


b1b2 minus λ2μ21Yμ21Z1113872 1113873ωe


ασ2ω

eminus r(Tminus t)


ωb21113888 1113889 t1 lt tle t7

ασ2ω

eminus r(Tminus t)

1 11113874 1113875 t7 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(100)


V(t x l)

1113957Q1(t x l) + 1113957Q5(t) + R4 0le tle t1


t7 lt tleT

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(101)

where

1113957Q10(t) a22η22


λa2η2μ1Yμ1Z

b2re

r(Tminus t)minus 11113872 1113873

1113957Q11(t) ωλ2μ21Yμ

21Z

4b2rminusωb1

4r1113888 1113889 e2r(Tminus t)( 1113857



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(102)


plowast1 (t) q

lowast1 (t) q

lowast2 (t)( 1113857

ασ2ω

eminus r(Tminus t)

a2η2ωb2

eminus r(Tminus t)

01113888 1113889 0le tle t8

ασ2ω

eminus r(Tminus t)

1 01113874 1113875 t8 lt tleT

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(103)


1113957Q1(t x l) + 1113957Q12(t) + R6 0le tle t8


⎧⎨

⎩ (104)

where

1113957Q12(t) a21η21

2ωb1(T minus t)


4re2r(Tminus t)

minus 11113872 1113873

1113957R6 1113957Q8(t) + 1113957Q13(t) minus 1113957Q12(t)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(105)





21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References
































21 and b2 2(λ + λ2)ξ

22











0 02 04 06 08 1t

15

2

25

3

35

4

45

5

p1

ω = 05ω = 075ω = 1


0 05 1 15 2t

38

4

42

44

46

48

5

52

54

56

58

p1

β = 01β = 02β = 03



t 0 1 2 3 4 5 6 7 81113954p1 38978 46503 55481 66192 78971 94218 112407 134109 1600001113954q1 04213 05026 05997 07155 08536 10184 12150 14496 172941113954q2 06019 07181 08567 10221 12194 14548 17357 20708 24706







02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References




































02

025

03

035

04

045

05

055

q2

0 02 04 06 08 1t

ω = 05ω = 075ω = 1


t

012

014

016

018

02

022

024

026

028

03

032

q1

0 02 04 06 08 1

ω = 05ω = 075ω = 1


q1

024

026

028

03

032

034

036

038

0 05 1 15 2t

β = 01β = 02β = 03


q2

04

045

05

055

06

065

0 05 1 15 2t

β = 01β = 02β = 03






6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References


































6 Conclusion



Data Availability




02

025

03

035

04

045

05

055

06

q

0 2 4 6 8 10λ1

q1q2


03

035

04

045

05

055

06

q

0 2 4 6 8 10λ2

q1q2


0 2 4 6 8 10025

03

035

04

045

05

055

06

065

q

λq1q2



Acknowledgments


References































Acknowledgments


References































optimaltime-consistentinvestmentandreinsurance ... · young [7] ﬁrst investigate the proportional...

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