Setting observation : fix , generating paths

resampling :

claim amount :

inter-claim :

premium rate :

interest rate :

Convergence ofFig.2 shows that converges to a unique smooth function of as resampling size and number of sample paths go to infinity.

Fig.3 displays the behaviour of curves of various numberof sample paths around the maximum of .

Convergence of In Fig.4, we can observe the consistency of . Table.1lists standard deviation of for each sample size and number of sample paths.

Under the regularity conditions, we prove uniformconvergence of and consistency of .To prove consistency of , we use the followingtheorem.Theorem (Consistency of M-estimator) Assume the following three conditions,

• ,• ,

• .

Then, .

Proposition

Insurance company is exposed to uncertainties —when insured event occurs and how much the claim amount is. Lundberg (1903) modelled the fluctuation of the surplus of own company as .We suppose that the insurance company will refund an excess of the surplus ( ) over a previously determined barrier ( ), as the dividends. We define thesurplus process ( ) as follows:

What is the Optimal Dividend Barrier?To define Optimal Dividend Barrier, we define as the function aggregated expectation of present value of the dividends until ruin time:

We define the Optimal Dividend Barrier by maximizing :

Assume that we observe the data: where and are th claim amount and inter-claim between th claim and th claim, respectively.We obtain sample paths by resampling from the data,and define the estimators of and as follows.Definition

,

.

where denotes dividend amount for under one of the sample paths: denotes sample mean for all permutations of .

Nonparametric Estimation for Optimal Dividend Barrier based on Empirical Process

U(t)

b

V (u, b)

V (u, b) = E

"Z Tb

0e��tdDb(t)

#

Introduction

V (u, b)

ObjectivesOur objectives in this research are as follows.

• To propose estimators

• To derive asymptotic properties (Consistency, Asymptotic normality, …)• To perform simulation• To analyze real data

In this poster, we focus on the consistency and simulation.Firstly, we propose an estimator of , and definethe estimator of as its maximizer.Secondly, we prove an asymptotic properties of estimators.Lastly, we examine the convergence of estimators bysimulation.

V (u, b)

b⇤

Estimator

References• Frees, E. W. (1986). Nonparametric estimation of the

probability of ruin. Astin Bulletin, 16(S1), S81-S90.• Gerber, H. U., & Shiu, E. S. (1998). On the time value

of ruin. North American Actuarial Journal, 2(1), 48-72.• Gerber, H. U., & Shiu, E. S. (2006). On optimal

dividend strategies in the compound Poisson model. North American Actuarial Journal, 10(2), 76-93.

• Kosorok, M. R. Introduction to empirical processes and semiparametric inference. 2008.

Simulation

Future work

Conclusion

{(Xi,�iT ), i = 1, . . . , n}

c = 2

� = 0.1

• We proved consistency of .• We showed the convergence of estimators by

simulation.

b̂⇤n

• We derive asymptotic normality of .• We try to perform simulation of the convergence of

estimators with various cases.• We analyze real data.

b̂⇤n

U(t) = u+ ct� S(t), S(t) =PN(t)

i=1 Xi

Ub(t) = u+ c

Z t

0I{Ub(s) < b}ds� S(t)

V (u, b) b⇤

bVn(u, b) := En

"Z Tb

0e��tdDb,n(t)

#

b̂⇤nP! b⇤

{(Xi,�iT ), i = 1, . . . , n}Xi �iT

i i� 1i

dDb,n(t) (t, t+ dt]

En{1, 2, . . . , n}

bVn(u, b)

Consistencyb̂⇤n

b̂⇤n

M

Xii.i.d.⇠ Ex(1)

�iTi.i.d.⇠ Ex(1)

bVn(u, b)

b̂⇤n

u

0 t

U(t)

t

u

0

Ub(t)

b

bVn(u, b)

b̂⇤nb̂⇤n

Fig.1 Sample paths of and U(t) Ub(t)

Fig.2 Convergence of

Fig.3 The behaviour of around the maximizer

Fig.4 Convergence of

Table.1 Standard deviation of

n

(Xij ,�ij0T ) ; ij , ij0 2 {1, . . . , n}o

n

(Xij ,�ij0T ) ; ij , ij0 2 {1, . . . , N}o

N n

bVn(u, b)

bVn(u, b)

b̂⇤n

b̂⇤n

) d(bn, b⇤) ! 0bVn(u, b̂

⇤n) = sup

b2R+bVn(u, b)� oP (1)

supb2R+

|bVn(u, b)� V (u, b)|P! 0

d(b̂⇤n, b⇤)

P! 0

supb2R+

|bVn(u, b)� V (u, b)|P! 0

b⇤ := argmaxb2R+

V (u, b)

ˆb⇤n := argmaxb2R+

bVn(u, b)

8{bn} 2 R+, lim infn!1

V (u, bn) � V (u, b⇤)

dividend

bVn(u, b)

data pathresampling calculation

repeat for all permutationsample mean

100 10,000 100,000 1,000,000

100 0.4057 0.1361 0.0637 0.0280

1000 0.4134 0.1304 0.0619 0.0283

Atsunobu Oishi, Graduate School of Science and Technology, Keio University Hiroshi Shiraishi, Department of Mathematics, Keio University

bVn(u, b)

Ub(t)

NM

b