november 18, 2021

Indefinite linear-quadratic optimal control of

mean-field stochastic differential equation with jump

diffusion: an equivalent cost functional method ∗

Guangchen Wang†, Wencan Wang‡

November 18, 2021

Abstract: In this paper, we consider a linear-quadratic optimal control problem of mean-field

stochastic differential equation with jump diffusion, which is also called as an MF-LQJ problem.

Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to

deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost func-

tionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems.

With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are fur-

ther characterized. Optimal control of the indefinite MF-LQJ problem is represented as a state

feedback via solutions of Riccati equations. As a by-product, the method provides a new way

to prove the existence and uniqueness of solution to mean field forward-backward stochastic

differential equation with jump diffusion (MF-FBSDEJ, for short), where existing methods in

literature do not work. Some examples are provided to illustrate our results.

Keywords: Equivalent cost functional, Existence and uniqueness of solution to MF-FBSDEJ,

Indefinite MF-LQJ problem, Riccati equation, Stochastic Hamiltonian system.

Mathematics Subject Classification: 93E20, 60H10, 34K50

1 Introduction

In recent years, there is an increasing interest in mean-field control theory in mathematics,

engineering and finance. Comparing with classical stochastic optimal control, a new feature of

∗This work was supported by the National Natural Science Foundation of China under Grants 61925306,

61821004, and 11831010.†School of Control Science and Engineering, Shandong University, Jinan 250061, P.R. China, E-mail:

[email protected]‡Corresponding author, School of Control Science and Engineering, Shandong University, Jinan 250100, P.R.

China, E-mail: [email protected]

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this problem is that both objective functional and system dynamics involve states and controls

as well as their expected values. There are rich literatures on deriving necessary conditions

for optimality. See, for example, Andersson and Djehiche [1], Buckdahn et al. [5], Djehiche et

al. [12], Wang et al. [28]. Linear quadratic (LQ, for short) problems of mean-field type have

also been investigated. Yong [31] systematically studied an LQ problem of mean-field stochastic

differential equation (MF-SDE). Later on, Sun [24] and Li et al. [18] concerned open-loop and

closed-loop solvabilities for LQ problems of MF-SDE, respectively. Elliott et al. [13] dealt with

an LQ problem of MF-SDE with discrete time setting. Qi et al. [22] investigated stabilization

and control problems for linear MF-SDE under standard assumptions. Barreiro-Gomez et al.

[2, 3] considered LQ mean-field-type games.

It is well known that jump diffusion processes play an increasing role in describing stochastic

dynamical systems, due to its wide application in financial, economic and engineering problems.

For example, a geometric Brownian motion is usually used to model stock price, but it cannot

reflect discontinuous characteristics, which may be induced by large fluctuations. There are

various literatures on related topic. The interested readers may refer to Haadem et al. [14],

Øksendal and Sulem [21], Shen et al. [23] for more information. Haadem et al. [14] obtained

a maximum principle for jump diffusion process with infinite horizon and dealt an optimal

portfolio selection problem. Shen et al. [23] investigated stochastic maximum principle of mean

field jump diffusion process with delay and applied their results to a bicriteria mean-variance

portfolio selection problem. Øksendal and Sulem [21] systematically discussed optimal control,

optimal stopping, and impulse control of jump diffusion processes.

In this paper, a kind of indefinite LQ problem of mean field type jump diffusion process is

investigated. Indefinite LQ problems, first studied by Chen et al. [9], have received considerable

attention. Chen et al. [10] employed the method of completion of squares to study an indefinite

stochastic LQ problem. Huang and Yu [15] and Yu [32] proposed an equivalent cost functional

method to deal with stochastic LQ problem with indefinite weighting matrices. Ni et al. [19, 20]

considered indefinite LQ problems of discrete-time MF-SDE for an infinite horizon and a finite

horizon, respectively. Li et al. [16] studied an indefinite LQ problem of MF-SDE by introducing

a relax compensator. Wang et al. [27] concerned with uniform stabilization and social optimality

for general mean field LQ control systems, where state weight is not assumed with the definiteness

condition. Indefinite MF-LQJ problems, which are natural generalizations of those in [26] and

[25], have been not yet completely studied. Tang and Meng [26] investigated a definite MF-LQJ

problem in finite horizon and derived two Riccati equations by decoupling optimality system.

It was shown that under Assumption (S) given in Section 3 with S(t) ≡ 0, S(t) ≡ 0, these

two Riccati equations are uniquely solvable and a feedback representation for optimal control is

obtained. We point out that Assumption (S) is exactly the definite condition when we study an

MF-LQJ problem. The MF-LQJ problem reduces to an NC-LQJ problem if S(t) ≡ 0, S(t) ≡ 0,

where “NC” is the capital initials for “no cross”. Tang et al. [25] focused on open-loop and

2

closed-loop solvabilities of an MF-LQJ problem, which extended results in [24, 18]. However,

the solvabilities of related Riccati equations without Assumption (S) have not been specified.

Inspired by [15] and [32], we use an equivalent cost functional method to deal with an

indefinite MF-LQJ problem. As a preliminary result, we discuss a definite MF-LQJ problem.

As mentioned above, the results obtained in [26] are not applicable for solving this MF-LQJ

problem. We introduce an invertible linear transformation, which links the MF-LQJ problem

with the corresponding NC-LQJ problem. Combining the results in existing literature with this

linear transformation, we obtain an optimal control of the MF-LQJ problem under Assumption

(S). Then we introduce two auxiliary functions to construct equivalent cost functionals. The

original MF-LQJ problem with indefinite control weighting matrices is transformed into an

MF-LQJ problem under Assumption (S). In a word, we can investigate the indefinite MF-LQJ

problem by using this method.

Our paper distinguishes from existing literature in the following aspects. (i) An indefinite

MF-LQJ problem is discussed in this paper, which generalized the results in [26, 25, 16]. The

considered model could characterize more general problems and the jump diffusion item is impor-

tant in some controlled dynamics system. As we will see in Example 5.1, there is no equivalent

cost functional satisfying Assumption (S) if jump diffusion item disappears, which implies that

we can not construct an optimal control directly in terms of Riccati equations. We further

discussed the existence and uniqueness of solutions to the corresponding stochastic Hamilto-

nian system and Riccati equations without Assumption (S), which have not been considered in

[26, 25]. (ii) Compared with derivation of optimal control for the definite NC-LQJ problem in

[26], we derive an feedback control of “Problem MF” under Assumption (S) through a simple

calculation. Actually, we introduce an invertible linear transformation, which links MF-LQJ

problem with the corresponding NC-LQJ problem. (iii) Our results provide an alternative and

effective way to obtain the solvability of an MF-FBSDEJ, which does not satisfy classical con-

ditions in existing literature. In fact, when we consider two equivalent cost functionals with the

same control system, we can get the equivalence by an invertible linear transformation between

the corresponding stochastic Hamiltonian systems. We point out that the equivalence is existed

in a family of stochastic Hamiltonian systems. Therefore, we can prove the solvability of a more

general MF-FBSDEJ. Moreover, sometimes an MF-FBSDEJ may coincide with the stochastic

Hamiltonian system of an MF-LQJ problem, which implies that the solvability of MF-FBSDEJ

is actually the solvability of corresponding stochastic Hamiltonian system. Thus, in order to ob-

tain the unique solvability of an MF-FBSDEJ, we need only to find an equivalent cost functional

satisfying Assumption (S) of the related MF-LQJ problem. (iv) Relying on the equivalent cost

functional method, we can investigate the solvability of indefinite Riccati equations by virtue

of the solvability of positive definite Riccati equations. In fact, the original MF-LQJ problem

with indefinite control weighting matrices can be transformed into a definite MF-LQJ problem

by looking for a simpler and more flexible equivalent cost functional. And there exists an equiv-

3

alent relation between the corresponding Riccati equations. Similarly, we can get solvabilities

of Riccati equations with indefinite condition.

The rest of this paper is organized as follows. In Section 2, we formulate an MF-LQJ problem

and give some assumptions throughout this paper. Section 3 aims to study the MF-LQJ problem

under Assumption (S). We reduce a general MF-LQJ problem to an NC-LQJ problem via an

invertible linear transformation. In Section 4, we present our main results. We use the equivalent

cost functional method to study an MF-LQJ problem with indefinite weighting matrices. Section

5 gives several illustrative examples. Finally, in Section 6, we conclude this paper.

2 Problem formulation

Let Rn×m be an Euclidean space of all n×m real matrices with inner product 〈·, ·〉 being given

by 〈M,N〉 7→ tr(M>N), where the superscript > denotes the transpose of vectors or matrices.

The induced norm is given by |M | =√tr(M>M). In particular, we denote by Sn the set of all

n×n symmetric matrices. We mean by an n×n matrix N ≥ 0 that N is a nonnegative matrix.

Let T > 0 be a fixed time horizon and (Ω,F ,F,P) be a complete filtered probability space.

The filtration F ≡ Ftt≥0 is generated by the following two mutually independent processes,

augmented by all the P-null sets: a standard 1-dimensional Brownian motion Wt and a Poisson

random measure N(dt, dθ) on R+×Θ, where Θ ⊆ R \ 0 is a nonempty set, with compensator

N(dt, dθ) = ν(dθ)dt, such that N([0, t], A) = N([0, t], A) − N([0, t], A) is a martingale for all

A ∈ B(Θ) satisfying ν(A) < ∞. B(Θ) is the Borel σ-field generated by Θ. Here, ν(dθ) is a

σ-finite measure on (Θ,B(Θ)) satisfying∫

Θ(1∧ θ2)ν(dθ) <∞, which is called the characteristic

measure. Then N(dt, dθ) = N(dt, dθ) − ν(dθ)dt is the compensated Poisson random measure.

For any Euclidean space M , we introduce the following spaces:

L∞(0, T ;M) =u : [0, T ]→M |u(·) is a bounded function

;

L2F(0, T ;M) =

u : [0, T ]×Ω→M |u(·) is an F-adapted stochastic process such that E

[∫ T0 |u(t)|2dt

]<

∞

;

S2F(0, T ;M) =

u : [0, T ]×Ω→M |u(·) is an F-adapted cadlag process in L2

F(0, T ;M) such that

E[

supt∈[0,T ] |u(t)|2]<∞

;

L2ν(M) =

r : [0, T ]×Θ→M |r(·) is a deterministic function such that supt∈[0,T ]

∫Θ |r(t, θ)|

2ν(dθ) <

∞

;

L2F,ν(0, T ;M) =

r : [0, T ] × Θ × Ω → M |r(·) is an F-predictable stochastic process such that

E[∫ T

0

∫Θ |r(t, θ)|

2ν(dθ)dt]<∞

.

4

Consider a controlled linear MF-SDEJ

dXt =AtXt + AtE[Xt] +Btut + BtE[ut]

dt

+CtXt + CtE[Xt] +Dtut + DtE[ut]

dWt

+

∫Θ

Et,θXt− + Et,θE[Xt−] + Ft,θut + Ft,θE[ut]

N(dt, dθ), t ∈ [0, T ],

X0 = x ∈ Rn,

(2.1)

where At, At, Bt, Bt, Ct, Ct, Dt, Dt, Et,θ, Et,θ, Ft,θ, Ft,θ are given matrix valued deterministic func-

tions. In the above equation, u, valued in Rm, is a control process and X, valued in Rn, is the

corresponding state process. In this paper, an admissible control u is defined as a predictable

process such that u ∈ L2F(0, T ;Rm). The set of all admissible controls is denoted by U [0, T ]. We

introduce a cost functional

J [u] =1

2E〈GXT , XT 〉+ 〈GE[XT ],E[XT ]〉

+

∫ T

0

⟨(Qt St

S>t Rt

)(Xt

ut

),

(Xt

ut

)⟩dt

+

∫ T

0

⟨(Qt St

S>t Rt

)(E[Xt]

E[ut]

),

(E[Xt]

E[ut]

)⟩dt

,

(2.2)

where G, G are symmetric matrices and Qt, St, Rt, Qt, St, Rt are deterministic matrix-valued

functions with Qt = Q>t , Rt = R>t , Qt = Q>t , Rt = R>t . Our MF-LQJ problem is stated as

follows.

Problem MF: For any x ∈ Rn, find a u∗ ∈ U [0, T ] such that

J [u∗] = infu∈U [0,T ]

J [u]. (2.3)

Any u∗ ∈ U [0, T ] satisfying (2.3) is called an optimal control of Problem MF, and the corre-

sponding state process X∗ = X(x, u∗) is called an optimal state process. (X∗, u∗) is called an

optimal pair.

The following assumptions will be in force throughout this paper.

(H1) The coefficients of state equation satisfyA, A, C, C ∈ L∞(0, T ;Rn×n), B, B,D, D ∈ L∞(0, T ;Rn×m),

E, E ∈ L2ν(Rn×n), F, F ∈ L2

ν(Rn×m).

(H2) The weighting matrices in cost functional satisfyQ, Q ∈ L∞(0, T ; Sn), R, R ∈ L∞(0, T ;Sm),

S, S ∈ L∞(0, T ;Rn×m), G, G ∈ Sn.

We can show that under (H1), for any u ∈ U [0, T ], (2.1) admits a unique solution X = X(x, u) ∈S2F(0, T ;Rn).

5

3 MF-LQJ problem under standard conditions

In this section, we aim at studying Problem MF under some standard conditions. We introduce

an invertible linear transformation, which links MF-LQJ problem with the corresponding NC-

LQJ problem.

Assumption (S): For some α0 > 0,Rt, Rt + Rt ≥ α0I,Qt − StR−1

t S>t ≥ 0,

Qt + Qt − (St + St)(Rt + Rt)−1(St + St)

> ≥ 0, t ∈ [0, T ],

G,G+ G ≥ 0.

We introduce a stochastic Hamiltonian system related to Problem MF

dXt =AtXt + AtE[Xt] +Btut + BtE[ut]

dt

+CtXt + CtE[Xt] +Dtut + DtE[ut]

dWt

+

∫Θ

Et,θXt− + Et,θE[Xt−] + Ft,θut + Ft,θE[ut]

N(dt, dθ),

dYt = −A>t Yt + A>t E[Yt] + C>t Zt + C>t E[Zt] +

∫Θ

(E>t,θrt,θ + E>t,θE[rt,θ]

)ν(dθ)

+QtXt + QtE[Xt] + Stut + StE[ut]dt+ ZtdWt +

∫Θrt,θN(dt, dθ),

X0 = x, YT = GXT + GE[XT ],

Rtut + RtE[ut] + S>t Xt− + S>t E[Xt−] +B>t Yt− + B>t E[Yt−]

+D>t Zt + D>t E[Zt] +

∫Θ

(F>t,θrt,θ + F>t,θE[rt,θ]

)ν(dθ) = 0.

(3.1)

Using the method in [26], we decouple the above Hamiltonian system and derive Riccati equa-

tions associated with Problem MF

Pt + PtAt +A>t Pt + C>t PtCt +

∫ΘE>t,θPtEt,θν(dθ) +Qt

−(St + PtBt + C>t PtDt +

∫ΘE>t,θPtFt,θν(dθ)

)Σ−1

0t

·(S>t +B>t Pt +D>t PtCt +

∫ΘF>t,θPtEt,θν(dθ)

)= 0,

PT = G,

(3.2)

6

Πt + Πt(At + At) + (At + At)>Πt + (Ct + Ct)

>Pt(Ct + Ct)

+

∫Θ

(Et,θ + Et,θ)>Pt(Et,θ + Et,θ)ν(dθ) +Qt + Qt

−[(St + St) + Πt(Bt + Bt) + (Ct + Ct)

>Pt(Dt + Dt)

+

∫Θ

(Et,θ + Et,θ)>Pt(Ft,θ + Ft,θ)ν(dθ)

]Σ−1

1t

·[(St + St)

> + (Bt + Bt)>Πt + (Dt + Dt)

>Pt(Ct + Ct)

+

∫Θ

(Ft,θ + Ft,θ)>Pt(Et,θ + Et,θ)ν(dθ)

]= 0,

ΠT = G+ G,

(3.3)

where

Σ0t = Rt +D>t PtDt +

∫ΘF>t,θPtFt,θν(dθ),

Σ1t = Rt + Rt + (Dt + Dt)>Pt(Dt + Dt) +

∫Θ

(Ft,θ + Ft,θ)>Pt(Ft,θ + Ft,θ)ν(dθ).

Note that (3.1) is a coupled MF-FBSDEJ, where the coupling comes from the last relation

(which is essentially the maximum condition in the usual Pontryagin type maximum principle).

Different from an NC-LQJ problem, there are additional items 2〈Xt, Stut〉 and 2〈E[Xt], StE[ut]〉in cost functional (2.2). Next, we want to reduce Problem MF to an NC-LQJ problem. For this,

we introduce a controlled system

dXt =A1t(Xt − E[Xt]) +Btut + [A1t + A1t]E[Xt] + BtE[ut]

dt

+C1t(Xt − E[Xt]) +Dtut + [C1t + C1t]E[Xt] + DtE[ut]

dWt

+

∫Θ

E1t,θ(Xt− − E[Xt−]) + Ft,θut + [E1t,θ + E1t,θ]E[Xt−] + Ft,θE[ut]

N(dt, dθ),

X0 = x,

(3.4)

and a cost functional

J [u] =1

2E〈GXT , XT 〉+ 〈GE[XT ],E[XT ]〉

+

∫ T

0

⟨(Q1t 0n×m

0m×n Rt

)(Xt − E[Xt]

ut − E[ut]

),

(Xt − E[Xt]

ut − E[ut]

)⟩dt

+

∫ T

0

⟨(Q1t + Q1t 0n×m

0m×n Rt + Rt

)(E[Xt]

E[ut]

),

(E[Xt]

E[ut]

)⟩dt

,

(3.5)

7

whereA1t = At −BtR−1

t S>t ,

A1t + A1t = At + At − (Bt + Bt)(Rt + Rt)−1(St + St)

>,

C1t = Ct −DtR−1t S>t ,

C1t + C1t = Ct + Ct − (Dt + Dt)(Rt + Rt)−1(St + St)

>,

E1t,θ = Et,θ − Ft,θR−1t S>t ,

E1t,θ + E1t,θ = Et,θ + Et,θ − (Ft,θ + Ft,θ)(Rt + Rt)−1(St + St)

>,

Q1t = Qt − StR−1t S>t ,

Q1t + Q1t = Qt + Qt − (St + St)(Rt + Rt)−1(St + St)

>.

The corresponding NC-LQJ problem is stated as follows.

Problem NC: For any x ∈ Rn, find a u∗ ∈ U [0, T ] such that

J [u∗] = infu∈U [0,T ]

J [u]. (3.6)

Similar to Problem MF, we write the stochastic Hamiltonian system and Riccati equations

corresponding to Problem NC

dXt =A1tXt + A1tE[Xt] +Btut + BtE[ut]

dt

+C1tXt + C1tE[Xt] +Dtut + DtE[ut]

dWt

+

∫Θ

E1t,θXt− + E1t,θE[Xt−] + Ft,θut + Ft,θE[ut]

N(dt, dθ),

dYt = −A>1tYt + A>1tE[Yt] + C>1tZt + C>1tE[Zt] +

∫Θ

(E>1t,θrt,θ + E>1t,θE[rt,θ]

)ν(dθ)

+Q1tXt + Q1tE[Xt]dt+ ZtdWt +


X0 = x, YT = GXT + GE[XT ],

Rtut + RtE[ut] +B>t Yt− + B>t E[Yt−] +D>t Zt + D>t E[Zt]

+

∫Θ

(F>t,θ ˜rt,θ + F>t,θE[ ˜rt,θ]

)ν(dθ) = 0,

(3.7)

Pt + PtA1t +A>1tPt + C>1tPtC1t +

∫ΘE>1t,θPtE1t,θν(dθ) +Q1t

−(PtBt + C>1tPtDt +

∫ΘE>1t,θPtFt,θν(dθ)

)Σ−1

0t

·(B>t Pt +D>t PtC1t +

∫ΘF>t,θPtE1t,θν(dθ)

)= 0,

PT = G,

(3.8)

8

Πt + Πt

(A1t + A1t

)+(A1t + A1t

)>Πt +

(C1t + C1t

)>Pt(C1t + C1t

)+

∫Θ

(E1t,θ + E1t,θ)>Pt(E1t,θ + E1t,θ)ν(dθ) +Q1t + Q1t

−[Πt(Bt + Bt) + (C1t + C1t)

>Pt(Dt + Dt) +

∫Θ

(E1t,θ + E1t,θ)>Pt(Ft,θ + Ft,θ)ν(dθ)

]Σ−1

1t

·[(Bt + Bt)

>Πt + (Dt + Dt)>Pt(C1t + C1t) +

∫Θ

(Ft,θ + Ft,θ)>Pt(E1t,θ + E1t,θ)ν(dθ)

]= 0,

ΠT = G+ G,

(3.9)

where

Σ0t = Rt +D>t PtDt +


Σ1t = Rt + Rt + (Dt + Dt)>Pt(Dt + Dt) +

∫Θ

(Ft,θ + Ft,θ)>Pt(Ft,θ + Ft,θ)ν(dθ).

Lemma 3.1. Let Assumption (S) hold. For any two pairs (X,u) and (X, u), we introduce a

linear transformation(X − E[X]

u− E[u]

)=

(In×n 0n×m

−R−1S> Im×m

)(X − E[X]

u− E[u]

), (3.10)

(E[X]

E[u]

)=

(In×n 0n×m

−(R+ R)−1(S + S)> Im×m

)(E[X]

E[u]

). (3.11)

Then the following two statements are equivalent:

(i). (X,u) is an admissible (optimal) control of Problem MF.

(ii). (X, u) is an admissible (optimal) control of Problem NC.

Moreover, we have J [u] = J [u].

Proof. It follows from (3.10) and (3.11) that(X − E[X]

u− E[u]

)=

(In×n 0n×m

R−1S> Im×m

)(X − E[X]

u− E[u]

),

(E[X]

E[u]

)=

(In×n 0n×m

(R+ R)−1(S + S)> Im×m

)(E[X]

E[u]

).

Then linear transformation (3.10) with (3.11) is invertible. Through direct calculation, it is easy

to verify statement (i) is equivalent to statement (ii) , and thus J [u] = J [u].

The above lemma tells us that there exists some equivalent relationship between Problem MF

and Problem NC. We now analyze the relationship in terms of stochastic Hamiltonian system

and Riccati equations, respectively.

9

Lemma 3.2. Under Assumption (S), (X∗, u∗, Y, Z, r) is the solution of (3.7) if and only if

(X∗, u∗, Y, Z, r) is the solution of (3.1).

Proof. According to Lemma 3.1, it is not difficult to draw the conclusion.

Lemma 3.3. Under Assumption (S), we have

1. Riccati equations (3.8) and (3.2) are the same.

2. Riccati equations (3.9) and (3.3) are the same.

Proof. For simplicity of notations, we denote

G(At, Bt, Ct, Dt, Et,θ, Ft,θ;Qt, Rt, St;Pt)

= PtAt +A>t Pt + C>t PtCt +

∫ΘE>t,θPtEt,θν(dθ) +Qt

−(PtBt + C>t PtDt +

∫ΘE>t,θPtFt,θν(dθ) + St

)Σ−1

0t

·(B>t Pt +D>t PtCt +

∫ΘF>t,θPtEt,θν(dθ) + S>t

).

It is enough to proveG(At, Bt, Ct, Dt, Et,θ, Ft,θ;Qt, Rt, St;Pt) = G(A1t, Bt, C1t, Dt, E1t,θ, Ft,θ;Q1t, Rt,0;Pt).

We have

Σ−10t

(B>t Pt +D>t PtCt +

∫ΘF>t,θPtEt,θν(dθ) + S>t

)− Σ−1

0t

(B>t Pt +D>t PtC1t +


)= Σ−1

0t

(D>t PtDt +

∫ΘF>t,θPtFt,θν(dθ) +Rt

)R−1t S>t

= R−1t S>t .

Thus, we calculate

G(At, Bt, Ct, Dt, Et,θ, Ft,θ;Qt, Rt, St;Pt)−G(A1t, Bt, C1t, Dt, E1t,θ, Ft,θ;Q1t, Rt,0;Pt)

= PtBtR−1t S>t + StR

−1t B>t Pt + C>t PtDtR

−1t S>t + StR

−1t D>t PtC1t

+

∫ΘE>t,θPtFt,θR

−1t S>t ν(dθ) +

∫ΘStR

−1t F>t,θPt,θE1t,θν(dθ)

+ StR−1t S>t −

(St + PtBt + C>t PtDt +

∫ΘE>t,θPtFt,θν(dθ)

)R−1t S>t

− StR−1t



)= 0n×n.

Consequently, we complete the proof.

10

We cite a result in [26], which plays a role in deriving an optimal control of Problem MF

under standard condition.

Lemma 3.4. Let (H1)-(H2) and Assumption (S) hold, then Riccati equations (3.8) and (3.9)

admit unique solutions P and Π, respectively. Further, the optimal pair (X∗, u∗) of Problem NC

satisfies

u∗t =− Σ−10t



)(X∗t − E[X∗t ])

− Σ−11t

[(Bt + Bt)

>Πt + (Dt + Dt)>Pt(C1t + C1t)

+

∫Θ


]E[X∗t ],

dX∗t =A1tX

∗t + A1tE[X∗t ] +Btu

∗t + BtE[u∗t ]

dt

+C1tX

∗t + C1tE[X∗t ] +Dtu

∗t + DtE[u∗t ]

dWt

+

∫Θ

E1t,θX

∗t− + E1t,θE[X∗t−] + Ft,θu

∗t + Ft,θE[u∗t ]

N(dt, dθ),

X∗0 = x.

Defining

Y ∗t = Pt(X∗t − E[X∗t ]) + ΠtE[X∗t ],

Z∗t =

[PtC1t − PtDtΣ

−10t



)](X∗t − E[X∗t ])

+Pt(C1t + C1t)− Pt(Dt + Dt)Σ

−11t

[(Bt + Bt)


+

∫Θ


]E[X∗t ],

r∗t,θ =

[PtE1t,θ − PtFt,θΣ−1

0t



)](X∗t − E[X∗t ])

+Pt(E1t,θ + E1t,θ)− Pt(Ft,θ + Ft,θ)Σ

−11t

[(Bt + Bt)


+

∫Θ


]E[X∗t ],

the 5-tuple (X∗, u∗, Y ∗, Z∗, r∗) is the unique solution to MF-FBSDEJ (3.7). Moreover,

infu∈U [0,T ]

J [u] =1

2〈Π0x, x〉, ∀x ∈ Rn.

Using the above lemmas, we obtain a main result of this section.

Theorem 3.1. If (H1)-(H2) and Assumption (S) hold, then Riccati equations (3.2) and (3.3)

admit unique solutions P and Π, respectively. Further, the optimal pair (X∗, u∗) of Problem MF

11

satisfies

u∗t =− Σ−10t

(S>t +B>t Pt +D>t PtCt +


)(X∗t − E[X∗t ])

− Σ−11t

[(St + St)

> + (Bt + Bt)>Πt + (Dt + Dt)

>Pt(Ct + Ct)

+

∫Θ


]E[X∗t ],

dX∗t =AtX

∗t + AtE[X∗t ] +Btu

∗t + BtE[u∗t ]

dt

+CtX

∗t + CtE[X∗t ] +Dtu

∗t + DtE[u∗t ]

dWt

+

∫Θ

Et,θX

∗t− + Et,θE[X∗t−] + Ft,θu


N(dt, dθ),

X∗0 = x.

Defining

Y ∗t = Pt(X∗t − E[X∗t ]) + ΠtE[X∗t ],

Z∗t =

[PtCt − PtDtΣ

−10t



)](X∗t − E[X∗t ])

+Pt(Ct + Ct)− Pt(Dt + Dt)Σ

−11t

[(St + St)

> + (Bt + Bt)>Πt

+(Dt + Dt)>Pt(Ct + Ct) +

∫Θ


]E[X∗t ],

r∗t,θ =

[PtEt,θ − PtFt,θΣ−1

0t



)](X∗t − E[X∗t ])

+Pt(Et,θ + Et,θ)− Pt(Ft,θ + Ft,θ)Σ

−11t

[(St + St)

> + (Bt + Bt)>Πt

+ +(Dt + Dt)>Pt(Ct + Ct)

∫Θ


]E[X∗t ],

(3.12)


infu∈U [0,T ]

J [u] =1

2〈Π0x, x〉, ∀x ∈ Rn.

Proof. We only need to prove the 5-tuple (X∗, u∗, Y ∗, Z∗, r∗) is the unique solution to MF-

FBSDEJ (3.1). By the linear transformation introduced in Lemma 3.1 and the representation

12

of optimal control u∗ for Problem NC in Lemma 3.4, we get

u∗t = u∗t − E[u∗t ] + E[u∗t ]

=−R−1t S>t (X∗t − E[X∗t ]) + u∗t − E[u∗t ]− (Rt + Rt)

−1(St + St)>E[X∗t ] + E[u∗t ]

=−R−1t S>t (X∗t − E[X∗t ])− Σ−1

0t

(B>t Pt +D>t PtC1t

+


)(X∗t − E[X∗t ])− (Rt + Rt)

−1(St + St)>E[X∗t ]

− Σ−11t

[(Bt + Bt)


+

∫Θ


]E[X∗t ]

=− Σ−10t

[Σ0tR

−1t S>t +B>t Pt +D>t Pt(Ct −DtR

−1t S>t )

+

∫ΘF>t,θPt(Et,θ − Ft,θR−1

t S>t )ν(dθ)

](X∗t − E[X∗t ])

− Σ−11t

Σ1t(Rt + Rt)

−1(St + St)> + (Bt + Bt)

>Πt

+ (Dt + Dt)>Pt

[Ct + Ct − (Dt + Dt)(Rt + Rt)

−1(St + St)>]

+

∫Θ

(Ft,θ + Ft,θ)>Pt

[Et,θ + Et,θ − (Ft,θ + Ft,θ)(Rt + Rt)

−1(St + St)>]ν(dθ)

E[X∗t ]

=− Σ−10t

[S>t +B>t Pt +D>t PtCt +


](X∗t − E[X∗t ])

− Σ−11t

[(St + St)

> + (Bt + Bt)>Πt + (Dt + Dt)

>Pt(Ct + Ct)

+

∫Θ


]E[X∗t ].

13

Using Lemma 3.2 and the representation of Y ∗, Z∗, r∗ in Lemma 3.4, we have

Y ∗t = Y ∗t

=Pt(X∗t − E[X∗t ]) + ΠtE[X∗t ]

=Pt(X∗t − E[X∗t ]) + ΠtE[X∗t ],

Z∗t = Z∗t

=

[PtC1t − PtDtΣ

−10t



)](X∗t − E[X∗t ])

+Pt(C1t + C1t)− Pt(Dt + Dt)Σ

−11t

[(Bt + Bt)


+

∫Θ


]E[X∗t ]

=Pt(Ct −DtR

−1t S>t )− PtDtΣ

−10t

[B>t Pt +D>t Pt(Ct −DtR

−1t S>t )

+

∫ΘF>t,θPt(Et,θ − Ft,θR−1

t S>t )ν(dθ)

](X∗t − E[X∗t ])

+Pt

[Ct + Ct − (Dt + Dt)(Rt + Rt)

−1(St + St)>]− Pt(Dt + Dt)Σ

−11t

[(Bt + Bt)

>Πt

+ (Dt + Dt)>Pt

(Ct + Ct −

(Dt + Dt

) (Rt + Rt

)−1 (St + St

)>)+

∫Θ

(Ft,θ + Ft,θ

)>Pt

(Et,θ + Et,θ −

(Ft,θ + Ft,θ

) (Rt + Rt

)−1 (St + St

)>)ν(dθ)

]E[X∗t ]

=

[PtCt − PtDtΣ

−10t



)](X∗t − E[X∗t ])


−11t

[(St + St)

> + (Bt + Bt)>Πt


∫Θ


]E[X∗t ].

We can also obtain that r∗t,θ satisfies (3.12) similarly. Then the proof is completed.

4 Indefinite MF-LQJ problem

In the case that Assumption (S) does not hold true, it is possible that Problem MF is well-posed

and an optimal pair exists. In this section, we will apply the equivalent cost functional method

to deal with Problem MF without Assumption (S).

Definition 4.1. For a given controlled system, if there exist two cost functionals J and J

satisfying: for any admissible controls u1 and u2, J [u1] < J [u2] if and only if J [u1] < J [u2],

then we say J is equivalent to J .

Remark 4.1. The following two statements are equivalent:

14

1. Cost functional J is equivalent to J ;

2. For any admissible controls u1 and u2,

(a). J [u1] < J [u2] if and only if J [u1] < J [u2];

(b). J [u1] = J [u2] if and only if J [u1] = J [u2];

(c). J [u1] > J [u2] if and only if J [u1] > J [u2].

We denote

Φ =ϕ ∈ L∞(0, T ;Sn)|ϕ is a deterministic continuous differential function

.

For any H,K ∈ Φ, we define JHK [u] = J [u]− 12〈K0x, x〉. Applying Ito formula to 〈Ht(Xt −

E[Xt]), Xt − E[Xt]〉+ 〈KtE[Xt],E[Xt]〉 on the interval [0, T ], we derive

JHK [u]

=1

2E〈GHK(XT − E[XT ]), XT − E[XT ]〉+ 〈(GHK + GHK)E[XT ],E[XT ]〉

+

∫ T

0

⟨(QHKt SHKt

>

SHKt RHKt

)(Xt − E[Xt]

ut − E[ut]

),

(Xt − E[Xt]

ut − E[ut]

)⟩dt

+

∫ T

0

⟨(QHKt + QHKt (SHKt + SHKt )>

SHKt + SHKt RHKt + RHKt

)(E[Xt]

E[ut]

),

(E[Xt]

E[ut]

)⟩dt

,

where

QHKt = Qt + Ht +HtAt +A>t Ht + C>t HtCt +

∫ΘE>t,θHtEt,θν(dθ),

SHKt = St +HtBt + C>t HtDt +

∫ΘE>t,θHtFt,θν(dθ),

RHKt = Rt +D>t HtDt +

∫ΘF>t,θHtFt,θν(dθ),

GHK = G−HT ,

QHKt + QHKt = Qt + Qt + Kt +Kt(At + At) + (At + At)>Kt

+ (Ct + Ct)>Ht(Ct + Ct) +

∫Θ

(Et,θ + Et,θ)>Ht(Et,θ + Et,θ)ν(dθ),

SHKt + SHKt = St + St +Kt(Bt + Bt) + (Ct + Ct)>Ht(Dt + Dt)

+

∫Θ

(Et,θ + Et,θ)>Ht(Ft,θ + Ft,θ)ν(dθ),

RHKt + RHKt = Rt + Rt + (Dt + Dt)>Ht(Dt + Dt) +

∫Θ

(Ft,θ + Ft,θ)>Ht(Ft,θ + Ft,θ)ν(dθ),

GHK + GHK = G+ G−KT .

(4.1)

Since JHK [u] and J [u] differ by only a constant −12〈K0x, x〉, they are equivalent. In other

words, we get a family of equivalent cost functionals JHK [u], which includes the original cost

15

functional J [u] when H ≡ 0, K ≡ 0. For the sake of convenience, given any H,K ∈ Φ, we call

Problem MF “Problem JHK” if J [u] is replaced by JHK [u].

We write down stochastic Hamiltonian systems corresponding to Problem JHK

dXHKt =

AtX

HKt + AtE[XHK

t ] +BtuHKt + BtE[uHKt ]

dt

+CtX

HKt + CtE[XHK

t ] +DtuHKt + DtE[uHKt ]

dWt

+

∫Θ

Et,θX

HKt− + Et,θE[XHK

t− ] + Ft,θuHKt + Ft,θE[uHKt ]

N(dt, dθ),

dY HKt = −

A>t Y

HKt + A>t E[Y HK

t ] + C>t ZHKt + C>t E[ZHKt ]

+

∫ΘE>t,θr

HKt,θ ν(dθ) +

∫ΘE>t,θE[rHKt,θ ]ν(dθ) +QHKt XHK

t + QHKt E[XHKt ]

+SHKt uHKt + SHKt E[uHKt ]dt+ ZHKt dW +

∫ΘrHKt,θ N(dt, dθ),

XHK0 = x, Y HK

T = GHKXHKT + GHKE[XHK

T ],

SHKt>XHKt− + SHK

>t E[XHK

t− ] +B>t YHKt− + B>t E[Y HK

t− ] +D>t ZHKt + D>t E[ZHKt ]

+

∫ΘF>t,θr

HKt,θ ν(dθ) +

∫ΘF>t,θE[rHKt,θ ]ν(dθ) +RHKt uHKt + RHKt E[uHKt ] = 0.

(4.2)

We note that (3.1) coincides with (4.2) while H ≡ 0,K ≡ 0. The following lemma shows that

there exists an equivalent relationship among Hamiltonian systems (4.2).

Lemma 4.1. For all H,K ∈ Φ, the existence and uniqueness of solutions to Hamiltonian

systems (4.2) are equivalent.

Proof. We only need to prove that for all H,K ∈ Φ, the existence and uniqueness of so-

lutions to Hamiltonian systems (4.2) are equivalent to (3.1). For any given H,K ∈ Φ, if

(XHK , uHK , Y HK , ZHK , rHK) is a solution of (4.2), we define

Xt = XHKt ,

ut = uHKt ,

Yt = Y HKt +Ht(X

HKt − E[XHK

t ]) +KtE[XHKt ],

Zt = ZHKt +Ht(CtXHKt + CtE[XHK

t ] +DtuHKt + DtE[uHKt ]),

rt,θ = rHKt,θ +Ht(Et,θXHKt− + Et,θE[XHK

t− ] + Ft,θuHKt + Ft,θE[uHKt ]).

Then (X,u, Y, Z, r) is a solution of (3.1). Thus we complete the proof.

We write down Riccati equations related to Problem JHK :

16

PtHK

+ PHKt At +A>t PHKt + C>t P

HKt Ct +

∫ΘE>t,θP

HKt Et,θν(dθ) +QHKt

−(SHKt + PHKt Bt + C>t P

HKt Dt +

∫ΘE>t,θP

HKt Ft,θν(dθ)

)ΣHK

0t−1

·(SHKt

>+B>t P

HKt +D>t P

HKt Ct +

∫ΘF>t,θP

HKt Et,θν(dθ)

)= 0,

PHKT = GHK ,

(4.3)

and

ΠtHK

+ ΠHKt (At + At) + (At + At)

>ΠHKt + (Ct + Ct)

>PHKt (Ct + Ct)

+

∫Θ

(Et,θ + Et,θ)>PHKt (Et,θ + Et,θ)ν(dθ) +QHKt + QHKt

−[(SHKt + SHKt ) + ΠHK

t (Bt + Bt) + (Ct + Ct)>PHKt (Dt + Dt)

+

∫Θ

(Et,θ + Et,θ)>PHKt (Ft,θ + Ft,θ)ν(dθ)

]ΣHK

1t−1

·[(SHKt + SHKt )> + (Bt + Bt)

>ΠHKt + (Dt + Dt)

>PHKt (Ct + Ct)

+

∫Θ

(Ft,θ + Ft,θ)>PHKt (Et,θ + Et,θ)ν(dθ)

]= 0,

ΠHKT = GHK + GHK ,

(4.4)

where

ΣHK0t = RHKt +D>t P

HKt Dt +

∫ΘF>t,θP

HKt Ft,θν(dθ),

ΣHK1t = RHKt + RHKt + (Dt + Dt)

>PHKt (Dt + Dt) +

∫Θ

(Ft,θ + Ft,θ)>PHKt (Ft,θ + Ft,θ)ν(dθ).

Lemma 4.2. For any H,K ∈ Φ, the existence and uniqueness of solutions to Riccati equations

associated with Problem JHK are equivalent.

Proof. We only need to prove that for all H,K ∈ Φ, the existence and uniqueness of solutions

to Riccati equations (4.3) and (4.4) are equivalent to Riccati equations (3.2) and (3.3). For any

given H,K ∈ Φ, if PHK ,ΠHK are solutions of Riccati equations (4.3) and (4.4), respectively,

then we definePt = PHKt +Ht,

Πt = ΠHKt +Kt.

Through a straightforward calculation, we know that P,Π are solutions of Riccati equations

(3.2) and (3.3), respectively. Thus we complete the proof.

We now give two theorems which are our main results in this section.

17

Theorem 4.1. If there exist H, K ∈ Φ such that (QHKt , SHKt , RHKt , GHK), (QHKt , SHKt , RHKt , GHK)

satisfy Assumption (S), then Riccati equations (3.2) and (3.3) admit unique solutions P,Π, re-

spectively. Further, the optimal pair (X∗, u∗) of Problem MF satisfies

u∗t =− Σ−10t


∫ΘF>t,θP

HKt Et,θν(dθ)

)(X∗t − E[X∗t ])

− Σ−11t

[(St + St)

> + (Bt + Bt)>Πt + (Dt + Dt)

>Pt(Ct + Ct)

+

∫Θ


]E[X∗t ],

dX∗t =(AtX

∗t + AtE[X∗t ] +Btu

∗t + BtE[u∗t ]

)dt

+(CtX

∗t + CtE[X∗t ] +Dtu

∗t + DtE[u∗t ]

)dWt

+

∫Θ

Et,θX

∗t− + Et,θE[X∗t−] + Ft,θu


N(dt, dθ),

X∗0 = x.

(4.5)

Defining

Y ∗t = Pt(X∗t − E[X∗t ]) + ΠtE[X∗t ],

Z∗t =

[PtCt − PtDtΣ

−10t



)](X∗t − E[X∗t ])


−11t

[(St + St)

> + (Bt + Bt)>Πt


∫Θ


]E[X∗t ],

r∗t,θ =

[PtEt,θ − PtFt,θΣ−1

0t



)](X∗t − E[X∗t ])

+Pt(Et,θ + Et,θ)− Pt(Ft,θ + Ft,θ)Σ

−11t

[(St + St)

> + (Bt + Bt)>Πt

+ (Dt + Dt)>Pt(Ct + Ct) +

∫Θ


]E[X∗t ],

(4.6)


infu∈U [0,T ]

J [u] =1

2〈Π0x, x〉, ∀x ∈ Rn.

Proof. We now consider Problem J HK . For H, K ∈ Φ, according to Theorem 3.1, Riccati

equations (4.3) and (4.4) associated with Problem J HK admit unique solutions P HK ,ΠHK ,

respectively. By Lemma 4.2, Riccati equations (3.2) and (3.3) admit unique solutions P,Π,

respectively. Using Theorem 3.1, further we know that the optimal pair (XHK , uHK) of Problem

18

J HK satisfies

uHKt =− ΣHK0t

−1(SHKt

>+B>t P

HKt +D>t P

HKt Ct

+

∫ΘF>t,θP

HKt Et,θν(dθ)

)(XHKt − E[XHK

t ])

− ΣHK1t

−1[(SHKt + SHKt )> + (Bt + Bt)

>ΠHKt + (Dt + Dt)

>P HKt (Ct + Ct)

+

∫Θ

(Ft,θ + Ft,θ)>P HKt (Et,θ + Et,θ)ν(dθ)

]E[XHK

t ],

dXHKt =

(AtX

HKt + AtE[XHK

t ] +BtuHKt + BtE[uHKt ]

)dt

+(CtX

HKt + CtE[XHK

t ] +DtuHKt + DtE[uHKt ]

)dWt

+

∫Θ

Et,θX

HKt− + Et,θE[XHK

t− ] + Ft,θuHKt + Ft,θE[uHKt ]

N(dt, dθ),

XHK0 = x.

Defining

Y HKt = P HKt

(XHKt − E[XHK

t ])

+ ΠHKt E[XHK

t ],

ZHKt,θ =[P HKt Ct − P HKt DtΣ

HK0t

−1(SHKt

>+B>t P

HKt +D>t P

HKt Ct

+

∫ΘF>t,θP

HKt Et,θν(dθ)

)](XHKt − E[XHK

t ])

+[P HKt (Ct + Ct)− P HKt (Dt + Dt)Σ

HK1t

−1(

(SHKt + SHKt )> + (Bt + Bt)>ΠHK

t

+ (Dt + Dt)>P HKt (Ct + Ct) +

∫Θ


)]E[XHK ],

rHKt,θ =[P HKt Et,θ − P HKt Ft,θΣ

HK0t

−1(SHKt

>+B>t P

HKt +D>t P

HKt Ct

+

∫ΘF>t,θP

HKt Et,θν(dθ)

)](XHKt − E[XHK

t ])

+[P HKt (Et,θ + Et,θ)− P HKt (Ft,θ + Ft,θ)Σ

HK1t

−1(

(SHKt + SHKt )> + (Bt + Bt)>ΠHK

t

+ (Dt + Dt)>P HKt (Ct + Ct) +

∫Θ


)]E[XHK

t ],

the 5-tuple (XHK , uHK , Y HK , ZHK , rHK) is a solution to MF-FBSDEJ (4.2) corresponding to

Problem J HK . Since J HK [u] is equivalent to J [u], then (XHK , uHK) is also an optimal pair of

19

Problem MF. By Lemma 4.1, we define

X∗t = XHKt ,

u∗t = uHKt ,

Y ∗t = Y HKt + Ht(X

HKt − E[XHK

t ]) + KE[XHKt ],

Z∗t = ZHKt + Ht(CtXHKt + CtE[XHK

t ] +DtuHKt + DtE[uHKt ]),

r∗t,θ = rHKt,θ + Ht(Et,θXHKt− + Et,θE[XHK

t− ] + Ft,θuHKt + Ft,θE[uHKt ]),

then the 5-tuple (X∗, u∗, Y ∗, Z∗, r∗) is a solution to MF-FBSDEJ (3.1). Through a simple

calculation, we know that the optimal pair (X∗, u∗) satisfies equation (4.5) and (Y ∗, Z∗, r∗)

satisfies (4.6). Moreover, it follows from Theorem 3.1 that

infu∈U [0,T ]

J HK [u] =1

2〈ΠHK

0 x, x〉, ∀x ∈ Rn.

Thus we have

infu∈U [0,T ]

J [u] =1

2〈ΠHK

0 x, x〉+1

2〈K0x, x〉

=1

2〈Π0x, x〉, ∀x ∈ Rn.

The proof is completed.

Theorem 4.2. If Riccati equations (3.2) and (3.3) admit unique solutions P,Π, respectively, and

Rt+D>PtDt+

∫Θ F

>t,θPtFt,θν(dθ) > α1I, Rt+Rt+(Dt+Dt)

>Pt(Dt+Dt)+∫

Θ(Ft,θ+Ft,θ)>Pt(Ft,θ+

Ft,θ)ν(dθ) > α1I for some α1 > 0, then there exist H, K ∈ Φ such that (QHKt , SHKt , RHKt , GHK)

and (QHKt , SHKt , RHKt , GHK) satisfy Assumption (S).

20

Proof. We consider the equivalent cost functional JPΠ[u]. It is easy to verify

QPΠt = Qt + Pt + PtAt +A>t Pt + C>t PtCt +

∫ΘE>t,θPtEt,θν(dθ),

SPΠt = St +B>t Pt +D>t PtCt +

∫ΘF>t,θPtEt,θν(dθ),

RPΠt = Rt +D>t PtDt +


GPΠ = G− PT ,

QPΠt + QPΠ

t = Qt + Qt + Πt + Πt(At + At) + (At + At)>Πt

+ (Ct + Ct)>Pt(Ct + Ct) +

∫Θ

(Et,θ + Et,θ)>Pt(Et,θ + Et,θ)ν(dθ),

SPΠt + SPΠ

t = St + St + Πt(Bt + Bt) + (Ct + Ct)>Pt(Dt + Dt)

+

∫Θ

(Et,θ + Et,θ)>Pt(Ft,θ + Ft,θ)ν(dθ),

RPΠt + RPΠ

t = Rt + Rt + (Dt + Dt)>Pt(Dt + Dt) +

∫Θ

(Ft,θ + Ft,θ)>Pt(Ft,θ + Ft,θ)ν(dθ),

GPΠ + GPΠ = G+ G−ΠT ,

and (QPΠt , SPΠ

t , RPΠt , GPΠ), (QPΠ

t , SPΠt , RPΠ

t , GPΠ) satisfy Assumption (S).

5 Examples

In this section, we present four illustrative examples, where Assumption (S) does not hold true

for original optimal control problems. Example 5.1 shows that an optimal control exists even

though Assumption (S) does not hold true. In Example 5.2, it is difficult to prove the existence

and uniqueness of solutions to related Riccati equations. We use the equivalent cost functional

method to construct an MF-LQJ problem which satisfies Assumption (S) first, and then we

obtain an optimal control of the original stochastic control problem via solutions of Riccati

equations. We also give the existence and uniqueness of solutions for a family of MF-FBSDEJs

as a byproduct of our results. With the in-depth study of Example 5.2, we apply our results

to prove the existence and uniqueness of solution to an MF-FBSDEJ in Example 5.3, where

existing methods in literature do not work. In Example 5.4, we apply our results to solve an

asset-liability management problem and give some numerical solutions.

Example 5.1: Consider a 1-dimensional controlled MF-SDEJdXt = (Xt − E[Xt] + ut + E[ut]) dt+ (2ut − E[ut])dWt +

∫[1,+∞)

e−θE[ut]N(dt, dθ),

X0 = x,

(5.1)

21

with a cost functional

J [u] =1

2E

2|X(T )|2 − |E[X(T )]|2

+

∫ 1

0

(− 3|Xt|2 + 3|E[Xt]|2 − 4|ut|2 + 2|E[ut]|2

)dt

.

(5.2)

With the data, Assumption (S) does not hold. We write down the stochastic Hamiltonian system

dXt = (Xt − E[Xt] + ut + E[ut]) dt+ (2ut − E[ut])dWt +

∫[1,+∞)


dYt = −Yt − E[Yt]− 3Xt + 3E[Xt]

dt+ ZtdW +

∫[1,+∞)

rt,θN(dt, dθ),

X0 = x, Y (T ) = 2X(T )− E[X(T )],

− 4ut + 2E[ut] + Yt− + E[Yt−] + 2Zt − E[Zt] +

∫[1,+∞)

e−θE[rt,θ]ν(dθ) = 0.

(5.3)

The corresponding Riccati equations arePt + 2Pt − 3− P 2t

4Pt − 4= 0,

PT = 2,Πt −4Π2

t

−2 + Pt + δPt= 0,

ΠT = 1,

where δ =∫

[1,+∞) e−2θν(dθ) > 0. Solving them, we get

Pt = 2, Πt =δ

2T − 2t+ δ.

Note that − 4 + 4Pt = 4,

− 2 + Pt + δPt = 2δ.

It follows from Theorem 4.2 that the equivalent cost functional JPΠ[u] satisfies Assumption (S).

According to Theorem 4.1, the optimal pair (X,u) satisfiesut =− 1

2(Xt − E[Xt])−

1

2T − 2t+ δE[Xt],

dXt = (Xt − E[Xt] + ut + E[ut]) dt+ (2ut − E[ut])dWt +

∫[1,+∞)


X0 = x.

22

Defining

Yt = 2(Xt − E[Xt])−δ

2T − 2t+ δE[Xt],

Zt = − 2(Xt − E[Xt]) +2

2T − 2t+ δE[Xt],

rt,θ =2e−θ

2T − 2t+ δE[Xt],

the 5-tuple (X,u, Y, Z, r) is a solution to MF-FBSDEJ (5.3). Moreover,

infu∈U [0,T ]

J [u] =δ

2(2T + δ)x2, ∀x ∈ R.

Example 5.2: Consider a 1-dimensional controlled MF-SDEJdXt =

(2Xt − E[Xt] + ut

)dt+ 2utdWt +

∫Θ

(Et,θXt− + Et,θE[Xt−])N(dt, dθ), t ∈ [0, T ],

X0 = x,

(5.4)

with a cost functional

J [u] =1

2EαX2

T − (α+ 1)E[XT ]2 +

∫ T

0

(4E[Xt]

2 + 4E[Xt]E[ut] +Rtu2t + RtE[ut]

2)dt,

(5.5)

where

α >1

2(T + 1)2, Rt = (t+ 1)3 − 2(t+ 1)2, Rt = 1− (t+ 1)3.

Clearly, Assumption (S) does not hold. Now we introduce an equivalent cost functional

JH0K0 [u] satisfying Assumption (S). Recalling (4.1), we have

QHKt = Ht + 4Ht + δ1tHt, SHKt = Ht,

RHKt = (t+ 1)3 − 2(t+ 1)2 + 4Ht, GHK = α−HT ,

QHKt + QHKt = 4 + Kt + 2Kt + δ2tHt, SHKt + SHKt = 2 +Kt,

RHKt + RHKt = 1− 2(t+ 1)2 + 4Ht,

GHK + GHK = −1−KT , ∀H,K ∈ Φ,

where δ1t =∫

ΘE2t,θν(dθ) > 0, δ2t =

∫Θ(Et,θ + Et,θ)

2ν(dθ) > 0. In particular, if we define

H0t = 12(t+ 1)2,K0t = 1

1+(T−t) − 2, then

QH0K0t = (t+ 1) + 2(t+ 1)2 +

δ1(t+ 1)2

2, SH0K0

t =1

2(t+ 1)2,

RH0K0t = (t+ 1)3, GH0K0 = α− 1

2(T + 1)2,

QH0K0t + QH0K0

t =1

(1 + T − t)2+

2

1 + T − t+δ2(t+ 1)2

2,

SH0K0t + SH0K0

t =1

1 + (T − t),

RH0K0t + RH0K0

t = 1, GH0K0 + GH0K0 = 0.

23

It is easy to see that Assumption (S) holds true for JH0K0 [u]. Then Theorem 4.1 and Lemma

4.1 imply that for any H,K ∈ Φ, the MF-FBSDEJ

dXHKt =

(2XHK

t − E[XHKt ] + uHKt

)dt+ 2uHKt dWt

+

∫Θ

(Et,θXHKt− + Et,θE[XHK

t− ])N(dt, dθ),

dY HKt = −

(2Y HK

t − E[Y HKt ] +

∫Θ

(Et,θrHKt,θ + Et,θE[rHKt,θ ])ν(dθ)

+QHKt XHKt + QHKt E[XHK

t ] + SHKt uHKt + SHKt E[uHKt ])dt

+ ZHKt dWt +

∫ΘrHKt,θ N(dt, dθ),

XHK0 = x, Y HK

T = GHKXHKT + GHKE[XHK

T ],

SHK>

t XHKt− + SHK

>t E[XHK

t− ] + Y HKt− + 2ZHKt +RHKt uHKt + RHKt E[uHKt ] = 0

has a unique solution. Further, Theorem 4.1 implies that Riccati equationsPt + 4Pt + δ1tPt −P 2t

(t+ 1)3 − 2(t+ 1)2 + 4Pt= 0,

PT = α

and Πt + 2Πt + δ2tPt + 4− (Πt + 2)2

1− 2(t+ 1)2 + 4Pt= 0,

ΠT = −1

admit unique solutions P,Π, respectively. And the optimal pair (X∗, u∗) satisfiesu∗t =− Pt

(t+ 1)3 − 2(t+ 1)2 + 4Pt(X∗t − E[X∗t ])− Πt

1− 2(t+ 1)2 + 4PtE[X∗t ],

dX∗t =(

2X∗t − E[X∗t ] + u∗t

)dt+ 2u∗tdWt +

∫Θ

(Et,θX∗t− + Et,θE[X∗t−])N(dt, dθ),

X∗0 = x.

We remark that the well-posedness of MF-FBSDE, i.e., the jump diffusion item in MF-

FBSDEJ disappears, has been well studied (see [4, 6, 8, 7]). In detail, Bensoussan [4] derived the

existence and uniqueness of solution of MF-FBSDE under a monotonicity condition. Carmona

and Delarue [6] obtained the solvability of MF-FBSDE by a compactness argument and the

Schauder fixed point theorem under a bound condition. Carmona and Delarue [8] took advantage

of the convexity of the Hamiltonian to apply the continuation method, and proved the existence

and uniqueness of solution of MF-FBSDE. Carmona and Delarue [7] derived the solvability

results by using an approximation procedure under some convexity condition. Moreover, Li

and Min [17] investigated the existence and uniqueness of solution to MF-FBSDEJ under a

monotonicity condition, which extended the results in previous literature. Different from the

24

works above, our equivalent method provides an alternative way to solve MF-FBSDEJ. Specially,

the original stochastic Hamiltonian system is of form

dXt = (2Xt − E[Xt] + ut) dt+ 2utdWt +

∫Θ

(Et,θXt− + Et,θE[Xt−])N(dt, dθ),

dYt = −(

2Yt − E[Yt] +

∫Θ

(Et,θrt,θ + Et,θE[rt,θ])ν(dθ)

)dt

+ ZtdWt +


X0 = x, YT = αXT − (α− 1)E[XT ],

Yt + 2Zt +Rtut + RtE[ut] = 0.

(5.6)

Since Rt = (t+ 1)3 − 2(t+ 1)2, Rt + Rt = 1− 2(t+ 1)2, we can not derive an expression of the

optimal control process u from the last equation in (5.6). The monotonicity condition in [4, 17]

and the bounded condition in [6] fail. Moreover, Carmona and Delarue [8, 7] assumed that cost

functional satisfies some convex condition, which is not true in our setting, thus the methods

in [7] and [8] fail. We emphasize that our method is also effective in proving the solvability of

MF-FBSDEJ with a slightly general and complicated form. The following example provides a

better understanding on this issue.

Example 5.3: Consider an MF-FBSDEJ

dXt = (2Xt + E[Xt] + Yt + Zt) dt+ (Yt + Zt)dWt +

∫ΘEt,θE[Xt−]N(dt, dθ),

dYt =−

2Yt + E[Yt]−Xt + E[Xt] +

∫ΘEt,θE[rt,θ]ν(dθ)

dt

+ ZtdWt +


X(0) =x, YT = 2XT − E[XT ],

(5.7)

where X,Y, Z, r are 1-dimensional stochastic processes. We claim that (5.7) does not satisfy the

25

conditions in [4, 6, 17]. Indeed, we have

E[(X1t −X2t)

(2Y2t + E[Y2t]−X2t + E[X2t] +

∫ΘEt,θE[r2t,θ]ν(dθ)

−(

2Y1t + E[Y1t]−X1t + E[X1t] +

∫ΘEt,θE[r1t,θ]ν(dθ)

))]+ E

[(Y1t − Y2t)

(2X1t + E[X1t] + Y1t + Z1t − (2X2t + E[X2t] + Y2t + Z2t)

)]+ E

[(Z1t − Z2t)

(Y1t + Z1t − (Y2t + Z2t)

)]+ E

[∫Θ

(r1t,θ − r2t,θ)Et,θ

(E[X1t−]− E[X2t−]

)ν(dθ)

]= E[(X1t −X2t)

2]− E[X1t −X2t]2 + E[(Y1t − Y2t + Z1t − Z2t)

2]

≥ 0,

E[(X1T −X2T )

(2X1T − E[X1T ]− (2X2T − E[X2T ])

)]≥ E[(X1T −X2T )2].

It implies that the monotonicity condition in [4, 17] fails. (5.7) is a linear MF-FBSDEJ,

which does not satisfy the bounded condition in [6].

We now prove the existence and uniqueness of solution of MF-FBSDEJ (5.7) with the help

of our results. Consider an MF-LQJ problem with a 1-dimensional state equationdXt = (2Xt + E[Xt] + ut) dt+ utdWt +


X0 = x.

An admissible control u is a predictable process such that u ∈ L2F(0, T ;R). Introduce a cost

functional

J [u] =1

2E[2X2

T − E[XT ]2 +

∫ T

0

(−u2

t −X2t + E[Xt]

2)dt

]. (5.8)

The corresponding Hamiltonian system is

dXt = (2Xt + E[Xt] + ut) dt+ utdWt +


dYt = −(

2Yt + E[Yt]−Xt + E[Xt] +

∫ΘEt,θE[rt,θ]ν(dθ)

)dt

+ ZtdWt +


X0 = x, YT = 2XT − E[XT ],

− ut + Yt− + Zt = 0.

(5.9)

Different from (5.6), we can derive an explicit expression of the optimal control process u from

the last equation in (5.9). In fact, it is easy to see that stochastic Hamiltonian system (5.9) is

26

exactly MF-FBSDEJ (5.7) with ut = Yt +Zt. Note that the cost functional does not satisfy the

convex conditions in [8, 7].

For the above MF-LQJ problem, it is clear that Assumption (S) does not hold. Now we

introduce an equivalent cost functional JH0K0 [u] satisfying Assumption (S). Recalling (4.1), we

haveQHKt = −1 + Ht + 4Ht, S

HKt = Ht,

RHKt = −1 +Ht, GHK = 2−HT ,

QHKt + QHKt = Kt + 6Kt + δtHt, SHKt + SHKt = Kt,

RHKt + RHKt = −1 +Ht, GHK + GHK = 1−KT , ∀H,K ∈ Φ,

where δt =∫

Θ E2t,θν(dθ). In particular, if we define H0t = 2,K0t = 1, then

QH0K0t = 7, SH0K0

t = 2, RH0K0t = 1, GH0K0 = 0,

QH0K0t + QH0K0

t = 6 + 2δt, SH0K0t + SH0K0

t = 1,

RH0K0t + RH0K0

t = 1, GH0K0 + GH0K0 = 0.

With the data, Assumption (S) holds true for JH0K0 [u]. Then it follows from Theorem 4.1

that MF-FBSDEJ (5.9) admits a unique solution (X,u, Y, Z, r), and (X,Y, Z, r) is exactly the

solution of (5.7).

Example 5.4: Consider a financial market consisting of a bond and a stock, in which two

assets are trading continuously within the time horizon [0, T ]. The dynamics of the bond price

process S1t is governed by dS1t = rtS1tdt,

S10 = s1,

where rt is the interest rate of the bond. The dynamics of the stock price process S2t is governed

by dS2t = µtS2tdt+ σtS2tdWt,

S20 = s2,

where µt and σt are the appreciation rate and the volatility coefficient of the stock, respec-

tively. For simplicity, we assume that the coefficients µt ≥ rt > 0, σt and 1σt

are bounded and

deterministic functions.

We assume that the trading of shares takes place continuously in a self-financing fashion and

there are no transaction costs. We denote by Nt the asset of an investor and by ut the amount

allocated in the stock share at time t. Clearly, the amount invested in the risk-free asset is

Nt − ut. Without liability, the asset of the investor Nt, evolves asdNt = [rtNt + (µt − rt)ut]dt+ σtutdWt,

N0 = n0.(5.10)

27

The investor’s accumulative liability at time t is denoted by Lt. Chiu and Li [11], Wei and Wang

[30] described the liability process by a geometric Brownian motion. In fact, it is possible that

the control strategy and the mean of asset of the investor can influence the liability process, due

to the complexity of the financial market and the risk aversion behavior of the investor. Such

an example can be found in Wang et al. [29], where the liability process depends on a control

strategy (for example, capital injection or withdrawal) of the firm. Along this line, we proceed

to improve the liability process here. Suppose that the dynamics of Lt satisfiesdLt =

(atLt + ctE[Nt]

)dt+ btLtdWt,

L0 = l0,(5.11)

where at is the appreciation rate of the liability and bt is the corresponding volatility which

satisfies the non-degeneracy condition. at, bt, ct are deterministic continuous functions on [0, T ].

Taking the liability into consideration, the SDE for the net wealth of the investor at time t,

denote by It, is obtained by subtracting (5.11) from (5.10),

dLt =(atLt + ctE[It] + ctE[Lt]

)dt+ btLtdWt,

L0 = l0,

dIt =(rtIt +

(rt − at

)Lt +

(µt − rt

)ut − ctE[It]− ctE[Lt]

)dt+

(σtut − btLt

)dWt,

I0 = n0 − l0.

(5.12)

Definition 5.1. An R-valued portfolio strategy u is called admissible, if u is F-adapted and

E[∫ T

0 u2tdt]<∞. The set of all admissible portfolio strategies is denoted by Uad.

For any u ∈ Uad, (5.12) admits a unique solution (L, I) ∈ S2F(0, T ;R2). We introduce a

performance functional of the investor, which is in the form of

J [u] = E∫ T

0

[L2t + (It − E[It])

2]dt+ (IT − E[IT ])2

.

Now we pose an asset-liability management problem as follows.

Problem AL: Find a portfolio strategy u∗ ∈ Uad such that

J [u∗] = infu∈Uad

J [u]. (5.13)

The problem implies that the investor aims to minimize the risk of the net wealth and the

liability over the whole time horizon, simultaneously.

It is easy to see that Problem AL is a special case of Problem MF. Denoting Xt = (Lt, It)>,

28

and x = (l0, n0 − l0)>, we have

At =

(at 0

rt − at rt

), At =

(ct ct

−ct −ct

), Bt =

(0

µt − rt

), C =

(bt 0

−bt 0

),

D =

(0

σt

), Q =

(1 0

0 1

), Q =

(0 0

0 −1

), G =

(0 0

0 1

), G =

(0 0

0 −1

),

B = 02×1, C = 02×2, D = 02×1, R = R = 0, S = S = 02×1.

Clearly, Assumption (S) does not hold. Define Ht =

(0 0

0 λt

),Kt = 02×2, where λt is the

solution of λt =

[(µt − rtσt

)2

− 2rt

]λt +

[rt − at +

bt(µt − rt)σt

]2

λ2t ,

λT =1

2.

Recalling (4.1), we have

QHKt =

(1 + b2tλt (rt − at)λt

(rt − at)λt 1 + 2rtλt + λt

), SHKt =

(−btσtλt

(µt − rt)λt

),

RHKt = λtσ2t , G

HK =

(0 0

0 12

), QHKt + QHKt =

(1 + b2tλt 0

0 0

),

SHKt + SHKt =

(−btσtλt

0

), RHKt + RHKt = λtσ

2t , G

HK + GHK = 0.

It is easy to see that Assumption (S) holds true for JHK [u]. According to Theorem 4.1, we know

that the following Riccati equations have unique solutionsPt + PtAt +A>t Pt + C>t PtCt +Qt − (PtBt + C>t PtDt)Σ

−1t (B>t Pt +D>t PtCt) = 0,

PT = G,(5.14)

Πt + Πt(At + At) + (At + At)

>Πt + C>t PtCt + (Qt + Qt)

− [ΠtBt + C>t PtDt]Σ−1t [B>t Πt +D>t PtCt] = 0,

ΠT = G+ G,

(5.15)

where

Σt = D>t PtDt.

An optimal portfolio strategy u∗ is given by

u∗t =− Σ−1t (B>t Pt +D>t PtCt)(Xt − E[Xt])− Σ−1

t [B>t Πt +D>t PtCt]E[Xt], (5.16)

29

where

dXt =[

At −BtΣ−1t (B>t Pt +D>t PtCt)

](Xt − E[Xt])

+[At + At −BtΣ−1

t (B>t Πt +D>t PtCt)]E[Xt]

dt

+[

Ct −DtΣ−1t (B>t Pt +D>t PtCt)

](Xt − E[Xt])

+[Ct −DtΣ

−1t (B>t Πt + (D>t PtCt)

]E[Xt]

dWt,

X0 = x.

Note that it is hard to give a more explicit expression of u∗t due to the complexity of Ric-

cati equations (5.14) and (5.15). We will use numerical simulation to illustrate the optimal

investment strategy u∗t and to analyse the relationship between u∗t and some common financial

parameters in our model. Here we only analyse the relationship between u∗t and the risk-free rate

rt, as well as the appreciation rate of the liability at, respectively. In the following discussion,

we fix T = 1, n0 = 1, l0 = 0.5, µt = 0.3, σt = 0.5, bt = 0.3, ct = 0.5.

Figure 1: The relationship between r and u∗t .

A.: The relationship between r and u∗t .

Let at = 0.2, [r1t, r2t, r3t, r4t] = [0.05, 0.1, 0.15, 0.2]. We plot Fig. 5.1 which illustrates the

relationship between the optimal investment strategy u∗t and the risk-free rate rt. From Fig.

5.1, we find that the higher the risk-free rate is, the higher the amount of investor’s wealth

allocated in the stock share is. It is reasonable that the net wealth expectation E[It] increases

30

Figure 2: The relationship between a and u∗t .

with the increase of the risk-free rate, which leads to the increase of the liability Lt. Thus, the

investor has to invest more in stock share to avoid the risk of the net wealth.

B: The relationship between a and u∗t .

Let rt = 0.05, [a1t, a2t, a3t, a4t] = [0.1, 0.2, 0.3, 0.4]. We plot Fig. 5.2 which illustrates the

relationship between the optimal investment strategy u∗t and the appreciation rate of the liability

at. Fig. 5.2 shows that the higher the appreciation rate of the liability is, the higher the amount

of investor’s wealth allocated in the stock share is. It is reasonable that the liability increases

with the increase of the appreciation rate of liability. Consequently, the investor has to invest

more in stock share to avoid the risk of the net wealth.

6 Concluding remarks

We use an equivalent cost functional method to deal with an indefinite MF-LQJ problem. Rely-

ing on this method, we transform an indefinite MF-LQJ problem to a definite MF-LQJ problem.

Compared with existing literature, we further consider the unique solvabilities of Riccati equa-

tions arising in indefinite MF-LQJ problems, which have not been studied before. Moreover,

the method provides an alternative approach to solve MF-FBSDEJ. Several illustrative exam-

ples show that our method is more effective than existing methods in proving the existence and

uniqueness of solution to MF-FBSDEJ. Our results are obtained with the framework of MF-

31

LQJ problem. We will further investigate some results for other stochastic system. For example,

LQ optimal control with delay has not been completely solved yet. We will try to extend this

method to the framework with delay.

References

[1] D. Andersson and B. Djehiche, “A maximum principle for SDEs of mean-field type,” Appl.

Math. Optim., vol. 63, no. 3, pp. 341-356, 2011.

[2] J. Barreiro-Gomez, T. E. Duncan, and H. Tembine, “Discrete-time linear-quadratic mean-

field-type repeated games: perfect, incomplete, and imperfect information,” Automatica,

vol. 112, 2020, Art. no. 108647.

[3] J. Barreiro-Gomez, T. E. Duncan, and H. Tembine, “Linear-quadratic mean-field-type

games: jump-diffusion process with regime switching,” IEEE Trans. Autom. Control, vol.

64, no. 10, pp. 4329-4336, 2019.

[4] A. Bensoussan, S. C. P. Yam, and Z. Zhang, “Well-posedness of mean-field type forward-

backward stochastic differential equations,” Stochastic Process. Appl., vol. 125, no. 9, pp.

3327-3354, 2015.

[5] R. Buckdahn, J. Li, and J. Ma, “A stochastic maximum principle for general mean-field

systems,” Appl. Math. Optim., vol. 74, no. 3, pp. 507-534, 2016.

[6] R. Carmona and F. Delarue, “Mean-field forward-backward stochastic diffierential equa-

tions,” Electron. Commun. Probab., vol. 18, no. 68, pp. 1-15, 2013.

[7] R. Carmona and F. Delarue, “Probabilistic analysis of mean-field games,” SIAM J. Control

Optim., vol. 51, no. 4, pp. 2705-2734, 2013.

[8] R. Carmona and F. Delarue, “Forward-backward stochastic diffierential equations and con-

trolled mckean-vlasov dynamics,” Ann. Probab., vol. 43, no. 5, pp. 2647-2700, 2015.

[9] S. Chen, X. Li, and X. Zhou, “Stochastic linear quadratic regulators with indefinite control

weight costs,” SIAM J. Control Optim., vol. 36, no. 5, pp. 1685-1702, 1998.

[10] S. Chen and X. Zhou, “Stochastic linear quadratic regulators with indefinite control weight

costs. ii,” SIAM J. Control Optim., vol. 39, no. 4, pp. 1065-1081, 2000.

[11] M. C. Chiu and D. Li, “Asset and liability management under a continuoustime mean-

variance optimization framework,” Insurance Math. Econom., vol. 39, no. 3, pp. 330-355,

2006.

32

[12] B. Djehiche, H. Tembine, and R. Tempone, “A stochastic maximum principle for risk-

sensitive mean-field type control,” IEEE Trans. Autom. Control, vol. 60, no. 10, pp. 2640-

2649, 2015.

[13] R. Elliott, X. Li, and Y. Ni, “Discrete time mean-field stochastic linear-quadratic optimal

control problems,” Automatica, vol. 49, no. 11, pp. 3222-3233, 2013.

[14] S. Haadem, B. Øksendal, and F. Proske, “Maximum principles for jump diffusion processes

with infinite horizon,” Automatica, vol. 49, no. 7, pp. 2267-2275, 2013.

[15] J. Huang and Z. Yu, “Solvability of indefinite stochastic riccati equations and linear

quadratic optimal control problems,” Syst. Control Lett., vol. 68, pp. 68-75, 2014.

[16] N. Li, X. Li, and Z. Yu, “Indefinite mean-field type linear-quadratic stochastic optimal

control problems,” Automatica, vol. 122, 2020, Art. no. 109627.

[17] W. Li and H. Min, “Fully coupled mean-field FBSDEs with jumps and related optimal

control problems,” Optim. Control Appl. Meth., vol. 42, no. 1, pp. 305-329, 2021.

[18] X. Li, J. Sun, and J. Yong, “Mean-field stochastic linear quadratic optimal control problems:

closed-loop solvability,” Probab. Uncertain. Quant. Risk, vol. 1, 2016, Art. no. 2.

[19] Y. Ni, X. Li, and J. Zhang, “Indefinite mean-field stochastic linearquadratic optimal control:

from finite horizon to infinite horizon,” IEEE Trans. Autom. Control, vol. 61, no. 11, pp.

3269-3284, 2015.

[20] Y. Ni, J. Zhang, and X. Li, “Indefinite mean-field stochastic linear quadratic optimal con-

trol,” IEEE Trans. Autom. Control, vol. 60, no. 7, pp. 1786-1800, 2014.

[21] B. Øksendal and A. Sulem, “Applied stochastic control of jump diffusions.” Springer-Verlag,

New York, 2007.

[22] Q. Qi, H. Zhang, and Z. Wu, “Stabilization control for linear continuous-time mean-field

systems,” IEEE Trans. Autom. Control, vol. 64, no. 8, pp. 3461-3468, 2019.

[23] Y. Shen, Q. Meng, and P. Shi, “Maximum principle for mean-field jump-diffusion stochastic

delay differential equations and its application to finance,” Automatica, vol. 50, no. 6, pp.

1565-1579, 2014.

[24] J. Sun, “Mean-field stochastic linear quadratic optimal control problems: open-loop solv-

abilities,” ESAIM Control Optim. Calc. Var., vol. 23, no. 3, pp. 1099-1127, 2017.

[25] C. Tang , X. Li, and T. Huang, “Solvability for indefinite mean-field stochastic linear

quadratic optimal control with random jumps and its applications,” Optim. Control Appl.

Meth., vol. 41, no. 6, pp. 2320-2348, 2020.

33

[26] M. Tang and Q. Meng, “Linear-quadratic optimal control problems for mean-field stochastic

diffierential equations with jumps,” Asian J. Control, vol. 21, no. 2, pp. 809-823, 2019.

[27] B. Wang, H. Zhang, and J. Zhang, “Mean field linear-quadratic control: uniform stabiliza-

tion and social optimality,” Automatica, vol. 121, 2020, Art no. 109088.

[28] G. Wang, C. Zhang, and W. Zhang, “Stochastic maximum principle for mean-field type

optimal control under partial information,” IEEE Trans. Autom. Control, vol. 59, no. 2,

pp. 522-528, 2014.

[29] G. Wang, H. Xiao, and G. Xing, “An optimal control problem for mean-field forward-

backward stochastic differential equation with noisy observation,” Automatica, vol. 86, pp.

104-109, 2017.

[30] J. Wei and T. Wang, “Time-consistent mean-variance asset-liability management with ran-

dom coefficients,” Insurance Math. Econom., vol. 77, pp. 84-96, 2017.

[31] J. Yong, “Linear-quadratic optimal control problems for mean-field stochastic diffierential

equations,” SIAM J. Control Optim., vol. 51, no. 4, pp. 2809-2838, 2013.

[32] Z. Yu, “Equivalent cost functionals and stochastic linear quadratic optimal control prob-

lems,” ESAIM Control Optim. Calc. Var., vol. 19, no. 1, pp. 78-90, 2013.

34

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