# Optimal Expected Utility of Dividend Payments with Proportional Reinsurance under VaR Constraints and Stochastic Interest Rate

Authors: Yuzhen Wen; Chuancun Yin
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we consider the problem of maximizing the expected discounted utility of dividend payments for an insurance company taking into account the time value of ruin. We assume the preference of the insurer is of the CRRA form. The discounting factor is modeled as a geometric Brownian motion. We introduce the VaR control levels for the insurer to control its loss in reinsurance strategies. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategy. Finally, we provide some numerical examples to illustrate the results and analyze the VaR control levels on the optimal strategy.


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## Body

## 1. Introduction

In recent years, dividend optimization problems for insurance company have attracted extensive attention. The problem of optimal dividend was proposed by De Finetti in 1957. He suggested that a company would seek to find a strategy in order to maximize the accumulated value of expected discounted dividends up to the ruin time. Since then, many results on optimal dividend problems have been obtained. Some literatures on the optimal dividend include Avanzi et al. [1], Avanzi et al. [2], Avram et al. [3], Bayraktar et al. [4], Dai et al. [5], Dai et al. [6], Hubalek and Schachermayer [7], Li and Wu [8], Loeffen [9], Ng [10], Thonhauser and Albrecher [11], Yao et al. [12], and Yin and Wen [13].On the other hand, reinsurance can effectively reduce the risk, so reinsurance is an important way for an insurer to control its risk exposure. Many interesting results have been obtained under reinsurance strategy in recent years. See, for example, Azcue and Muler [14], who studied the optimal reinsurance in the framework of Cramér-Lundberg model. Zhou and Yuen [15] considered reinsurance for a diffusion model with capital injection under variance premium principle. For some related discussions, among others, we refer the reader to Li and Shen [16], Liang and Palmowski [17], Wen and Yin [18], Wu [19], and Yao et al. [20].In practice, Value-at-Risk (VaR) is often used to measure and control the risk of an insurance company. Recently, Chen et al. [21] investigated the optimal reinsurance strategies and the minimum probability of ruin with VaR constraints. Zhang et al. [22] discussed the optimal reinsurance strategies and the survival probability under dynamic VaR constraints. Bi and Cai [23] studied the equilibrium strategies under the mean-variance criterion with VaR constraints. Other works about optimal reinsurance problems with VaR constraints can be found in Cai and Tan [24], Cai et al. [25], Cheung et al. [26], and Yiu [27].The interest rate is the main factor of an insurance company. And, it is affected by many factors. So, we must pay attention to the current situation of interest rate and its changing trend. It is more reasonable to assume that the interest rate is a function of time. Eisenberg [28] solved the optimal dividends under a stochastic interest rate for the case of the Brownian motion. In this paper, we are going to study the problem of maximizing the expected discounted utility of dividend, taking into account both reinsurance under VaR constraints and a stochastic interest rate. The insurer’s surplus process is approximated by a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equation, we obtain the value function and the corresponding optimal strategies with and without VaR constraints.This paper is organized as follows. In Section2, we provide a general formulation of the optimal reinsurance-dividend problem. Then, we investigate the problem of the expected discounted utility of dividend maximization under noncheap reinsurance assumption. We give the explicit expressions for the expected discounted utility of dividend payments and the optimal strategies in Section 3. In Section 4, we consider the optimization problem with VaR constraints and solve the optimization problem using the results derived in Section 3. We also present some numerical examples to illustrate the results in Section 5.


## 2. Model Settings and Problem Formulations

LetΩ,F,P be a probability space equipped with a filtration Ftt≥0.


### 2.1. Reserve Process of an Insurer

The reserve processRtt≥0 of an insurer is modeled by
(1)dRt=1+θadt−dYt,R0=xfor the aggregate cumulative amount of claims counted up to time t:
(2)dYt=adt−bdWt,Y0=0,where a and b are positive constants, x≥0 is the initial surplus, 1+θa is the premium rate with the safety loading θ>0, and Wtt≥0 is a standard Brownian motion.In the following, the insurer is allowed to pay dividends. The cumulated dividends are described by a processC=Ctt≥0, which is an adapted and nondecreasing process. Ct represents the total dividends up to time t. In this paper, we assume that Ctt≥0 is absolutely continuous with respect to the Lebesgue measure. We suppose that the process C admits almost surely a density process denoted by ct≥0 modeling the intensity of the dividend payments in continuous time. To manage the risk, the insurer could choose a reinsurance policy qt at time t. When qt∈0,1, it means the insurer purchases proportional reinsurance. In this case, the insurer only pays its 1−qtYt, while the reinsurer pays the rest qtYt. A strategy π is said to be admissible, if the ruin cannot be caused by the dividend payment, and qt,ct is Ft-progressively measurable and satisfies 0≤qt≤1, ct≥0 for all t≥0. We denote the set of all admissible strategies by Π. The set of admissible strategies consists of nonnegative, nondecreasing, adapted, cádlág process. Let Xtπ denote the insurers total wealth at time t under the strategy πt=qt,ct. Then, the dynamic of Xtπ is given by
(3)dXtπ=1+θa−1−qtadt+b1−qtdWt−1+ηqtadt−ctd=θ−ηqtadt+b1−qtdWt−ctdt,t≥0,where η>θ is the safety loading of the reinsurer and the condition of η>θ is required for avoiding the insurers arbitrage.As a risk measure, we consider that dividends are discounted by the geometric Brownian motion(4)e−r−mt−δBt,ifm>δ22,where r>0, m>0, and δ≥0. Let further Btt≥0 be a standard Brownian motion independent of Wtt≥0. We suppose that the underlying filtration Ft is generated by Wt,Bt. The time of ruin for this process is defined by τπ=inft≥0:Xtπ≤0. We define the return function corresponding to π to be
(5)Vπr,x=Ex∫0τπe−r−ms−δBsucsds+∫0τπe−r−ms−δBsλds,where u is some fixed utility function, λ>0, Ex means expectation with respect to ℙx·=P·∣X0=x. Usually, it is assumed that u:ℝt≥0→ℝt≥0 is differentiable and nonnegative. e−βsλ can be interpreted as the present value of an amount which the insurer earns as long as the company is alive. In this way, the lifetime of the portfolio becomes part of the value function and is weighted according to the choice of λ. We seek to find an optimal strategy π∗ so that
(6)Vr,x=maxπ∈ΠVπr,x=Vπ∗r,x.In the present paper, we consider the Constant Relative Risk Aversion (CRRA) utility function:(7)uc=cpp,p∈0.1.


### 2.2. Value-at-Risk Constraints for the Proportional Reinsurance and Dividend Strategy

Under the reinsurance-dividend strategyπt=qt,ct, the insurer’s wealth process Xtπ,t≥0 is a risk process. The insurer may or has to use the risk measure of VaR to control its wealth for avoiding huge loss. For h>0 is small enough and for s in the interval t,t+h, we approximate πs by πt, s∈t,t+h. This is a reasonable approximation since the insurance strategy can only be adjusted at discrete time, and the decision is made based on the surplus at time t. Thus, the loss of the insurer in time interval t,t+h can be expressed as ΔXt,hπ≔Xtπ−Xt+hπ. Then, the SDE (3) admits a solution
(8)Xt+hπ=Xtπ+∫tt+hθ−ηqsads+∫tt+hb1−qsdWs−∫tt+hcsd=Xtπ+θ−ηqtah−cth+b1−qt∫tt+hdWs.Thus,(9)ΔXt,hπ=−θ−ηqtah+cth−b1−qt∫tt+hdWs.For a given risk levelk∈0,1 and a given horizon h>0, we denote the conditional VaR of ΔXt,hπ conditioning on Ft by VaRtk,h,π, namely,
(10)VaRtk,h,π≔infL∈ℝ;ℙΔXt,hπ≥L∣Ft≤k.Proposition 1.
Given risk levelk∈0,1 and time length h>0. We have
(11)VaRtk,h,π=cth−θ−ηqtah−Φ−1kb1−qth,where Φ−1·=infx∈ℝ:Φx≥k is the inverse function of the cumulative standard normal distribution function Φx=∫−∞x1/2πe−z2/2dz.Proof.
We have(12)ℙΔXt,hπ≥L∣Ft=ℙ−θ−ηqtah+cth−b1−qt∫tt+hdWs≥L∣Ft=ℙb1−qt∫tt+hdWs≤cth−θ−ηqtah−L∣Ft=ℙ1h∫tt+hdWs≤cth−θ−ηqtah−Lhb1−qt∣Ft=Φcth−θ−ηqtah−Lhb1−qt,where the last equality follows from the fact that 1/h∫tt+hdWs conditional on the filtration Ft is a standard normal random variable. Then, we have
(13)ℙΔXt,hπ≥L∣Ft≤k⇔Φcth−θ−ηqtah−Lhb1−qt≤k⇔L≥cth−θ−ηqtah−Φ−1kb1−qth.
Thus, we obtain (11).In this paper, we will derive the optimal strategy under the constraint(14)VaRtk,h,π≤VaR¯,t≥0.


## 2.1. Reserve Process of an Insurer

The reserve processRtt≥0 of an insurer is modeled by
(1)dRt=1+θadt−dYt,R0=xfor the aggregate cumulative amount of claims counted up to time t:
(2)dYt=adt−bdWt,Y0=0,where a and b are positive constants, x≥0 is the initial surplus, 1+θa is the premium rate with the safety loading θ>0, and Wtt≥0 is a standard Brownian motion.In the following, the insurer is allowed to pay dividends. The cumulated dividends are described by a processC=Ctt≥0, which is an adapted and nondecreasing process. Ct represents the total dividends up to time t. In this paper, we assume that Ctt≥0 is absolutely continuous with respect to the Lebesgue measure. We suppose that the process C admits almost surely a density process denoted by ct≥0 modeling the intensity of the dividend payments in continuous time. To manage the risk, the insurer could choose a reinsurance policy qt at time t. When qt∈0,1, it means the insurer purchases proportional reinsurance. In this case, the insurer only pays its 1−qtYt, while the reinsurer pays the rest qtYt. A strategy π is said to be admissible, if the ruin cannot be caused by the dividend payment, and qt,ct is Ft-progressively measurable and satisfies 0≤qt≤1, ct≥0 for all t≥0. We denote the set of all admissible strategies by Π. The set of admissible strategies consists of nonnegative, nondecreasing, adapted, cádlág process. Let Xtπ denote the insurers total wealth at time t under the strategy πt=qt,ct. Then, the dynamic of Xtπ is given by
(3)dXtπ=1+θa−1−qtadt+b1−qtdWt−1+ηqtadt−ctd=θ−ηqtadt+b1−qtdWt−ctdt,t≥0,where η>θ is the safety loading of the reinsurer and the condition of η>θ is required for avoiding the insurers arbitrage.As a risk measure, we consider that dividends are discounted by the geometric Brownian motion(4)e−r−mt−δBt,ifm>δ22,where r>0, m>0, and δ≥0. Let further Btt≥0 be a standard Brownian motion independent of Wtt≥0. We suppose that the underlying filtration Ft is generated by Wt,Bt. The time of ruin for this process is defined by τπ=inft≥0:Xtπ≤0. We define the return function corresponding to π to be
(5)Vπr,x=Ex∫0τπe−r−ms−δBsucsds+∫0τπe−r−ms−δBsλds,where u is some fixed utility function, λ>0, Ex means expectation with respect to ℙx·=P·∣X0=x. Usually, it is assumed that u:ℝt≥0→ℝt≥0 is differentiable and nonnegative. e−βsλ can be interpreted as the present value of an amount which the insurer earns as long as the company is alive. In this way, the lifetime of the portfolio becomes part of the value function and is weighted according to the choice of λ. We seek to find an optimal strategy π∗ so that
(6)Vr,x=maxπ∈ΠVπr,x=Vπ∗r,x.In the present paper, we consider the Constant Relative Risk Aversion (CRRA) utility function:(7)uc=cpp,p∈0.1.


## 2.2. Value-at-Risk Constraints for the Proportional Reinsurance and Dividend Strategy

Under the reinsurance-dividend strategyπt=qt,ct, the insurer’s wealth process Xtπ,t≥0 is a risk process. The insurer may or has to use the risk measure of VaR to control its wealth for avoiding huge loss. For h>0 is small enough and for s in the interval t,t+h, we approximate πs by πt, s∈t,t+h. This is a reasonable approximation since the insurance strategy can only be adjusted at discrete time, and the decision is made based on the surplus at time t. Thus, the loss of the insurer in time interval t,t+h can be expressed as ΔXt,hπ≔Xtπ−Xt+hπ. Then, the SDE (3) admits a solution
(8)Xt+hπ=Xtπ+∫tt+hθ−ηqsads+∫tt+hb1−qsdWs−∫tt+hcsd=Xtπ+θ−ηqtah−cth+b1−qt∫tt+hdWs.Thus,(9)ΔXt,hπ=−θ−ηqtah+cth−b1−qt∫tt+hdWs.For a given risk levelk∈0,1 and a given horizon h>0, we denote the conditional VaR of ΔXt,hπ conditioning on Ft by VaRtk,h,π, namely,
(10)VaRtk,h,π≔infL∈ℝ;ℙΔXt,hπ≥L∣Ft≤k.Proposition 1.
Given risk levelk∈0,1 and time length h>0. We have
(11)VaRtk,h,π=cth−θ−ηqtah−Φ−1kb1−qth,where Φ−1·=infx∈ℝ:Φx≥k is the inverse function of the cumulative standard normal distribution function Φx=∫−∞x1/2πe−z2/2dz.Proof.
We have(12)ℙΔXt,hπ≥L∣Ft=ℙ−θ−ηqtah+cth−b1−qt∫tt+hdWs≥L∣Ft=ℙb1−qt∫tt+hdWs≤cth−θ−ηqtah−L∣Ft=ℙ1h∫tt+hdWs≤cth−θ−ηqtah−Lhb1−qt∣Ft=Φcth−θ−ηqtah−Lhb1−qt,where the last equality follows from the fact that 1/h∫tt+hdWs conditional on the filtration Ft is a standard normal random variable. Then, we have
(13)ℙΔXt,hπ≥L∣Ft≤k⇔Φcth−θ−ηqtah−Lhb1−qt≤k⇔L≥cth−θ−ηqtah−Φ−1kb1−qth.
Thus, we obtain (11).In this paper, we will derive the optimal strategy under the constraint(14)VaRtk,h,π≤VaR¯,t≥0.


## 3. Solution to the Optimization Problem without VaR Constraints

In this section, we will solve the optimal reinsurance-dividend problem without VaR constraints. The HJB equation corresponding to the problem (6) is given by
(15)max0≤q≤1,c≥0θ−ηqa−cVx+12b21−q2Vxx+mVr+δ22Vrr+e−ruc+λ=0,with the boundary condition
(16)V0,0=0.We conjecture that the value function can be written in the formVr,x=e−rFx. In this case, the HJB equation becomes
(17)max0≤q≤1,c≥0θ−ηqa−cF′x+12b21−q2F′′x+δ22−mFx+uc+λ=0.Note that supremum in (17) without constraints is realized for
(18)qx=1+ηab2F′xF′′x,cx=F′x−11−p.Denoteα=1+2b2m−δ2/2/η2a2. Let
(19)B=−2b2η2a21−p1−α+αp>0,(20)D=2b2αη2aη−θ>0.We will solve our optimization problem under the assumption:Assumption 2.
For noncheap reinsurance, assume thatη>θ, α1−p>1 and 2−α1−pη−2θ<0.For the optimal value functionVr,x, we can derive the following property.Lemma 3.
The optimal value functionVr,x satisfies Vx≥0, Vr≤0, and Vxx<0.Proof.
It is easy to see thatVr,x is increasing in x and decreasing in r. We prove the property of Vxx<0. For 0<y<x, denote τyπ=inft:Xtπ=y. From the dynamic programming principle, we claim that Vr,x satisfies the following equality:
(21)Vr,x=maxπ∈ΠEx∫0τyπe−r−ms−δBsucs+λds+e−mτyπVr,y.
We first show the≤ part. Let ε>0. There exists π~∈Π such that Vπ~r,x>Vr,x−ε. We choose the strategy π¯=π1s∈0,τy+π~1s∈τy,τ where π is an admissible strategy. Then,
(22)Vr,x≥Vπ¯r,x=Ex∫0τyπe−r−ms−δBsucs+λd+∫τyπτπe−r−ms−δBsucs+λds=Ex∫0τyπe−r−ms−δBsucs+λds+∫0τπ−τyπe−r−ms+τyπ−δBs+τyπucs+τyπ+λds=Ex∫0τyπe−r−ms−δBsucs+λds+e−mτyπVπ~r,y≥Ex∫0τyπe−r−ms−δBsucs+λds+e−mτyπVr,y−ε,and therefore, we have
(23)Vr,x≥maxπ∈ΠEx∫0τyπe−r−ms−δBsucs+λds+e−mτyπVr,y.
For the other direction, since(24)Vπr,x=Ex∫0τyπe−r−ms−δBsucs+λds+e−mτyπVπr,y≤Ex∫0τyπe−r−ms−δBsucs+λds+e−mτyπVr,y,we have
(25)Vr,x≤maxπ∈ΠEx∫0τyπe−r−ms−δBsucs+λds+e−mτyπVr,y.
Thus, (21) is proved.
Forh>0, let Πh be the set of strategies π such that
(26)∫0ζθ−ηqsads+∫0ζb1−qsdWs−∫0ζcsds=−h,on the set ζ<∞, where ζ is a stopping time defined by
(27)ζ=inft≥0:∫0tθ−ηqsads+∫0tb1−qsdWs−∫0tcsds=−h.
By puttingh=x−y from (21), we obtain that
(28)Vr,x=maxπ∈ΠEx∫0ζe−r−ms−δBsucs+λds+e−mζVr,y.
Then,(29)Vr,x−Vr,x−h=maxπ∈ΠEx∫0ζe−r−ms−δBsucs+λds+e−mζ−1Vr,x−h=maxπ∈ΠEx∫0ζe−r−ms−δBsucs+λds+Exe−mζ−1Vr,x−h.
SinceExe−mζ<1 and Vr,x is a nondecreasing function of x, the right-hand side of (29) is a decreasing function of x. Thus, Vr,x−Vr,x−h is decreasing in x. Hence, Vx is also decreasing and Vxx<0. This finishes the proof.Proposition 4.
If Assumption2 is true, then on 0,x∗, the function v solving (15) is given by
(30)vr,x=e−rm−δ2/21α−1η−θa+1−p21−αp1−α+αpeξ1−pe−ξ+λ,where B and D are given in (19) and (20) and ξ=g−1x, where g−1 is the inverse of function g:
(31)gξ=1−pBeξ1−p+Dξ+Q1,for Q1=−1−pBeξ0/1−p−Dξ0 and ξ0 satisfies
(32)e−ξ01−ααη−θa+1−p21−αp1−α−αpepξ01−p+λ=0.Above(33)x∗=gξ∗,where
(34)ξ∗=1−plnb2−ηaDηaB.The maximizersq∗x and c∗x are given by
(35)q∗x=1−ηab2g′ξ,c∗x=ervx−11−p.Proof.
From the expression ofqx=1+ηa/b2F′x/F′′x and cx=F′x−1/1−p, we have that the HJB equation of (17) becomes
(36)θ−ηaF′x−η2a22b2F′x2F′′x+δ22−mFx+1−ppF′x−p1−p+λ=0.
Lettingx=gξ and F′gξ=e−ξ, then we have F′′gξ=−e−ξ/g′ξ. Plugging it into (36) and taking derivatives with ξ produce
(37)θ−ηae−ξ+η2a22b2g′ξe−ξ+δ22−mFgξ+1−ppep1−pξ+λ=0.
We take derivatives withξ, and we get
(38)η2a22b2g′′ξ−η2a22b2+m−δ22g′ξ−θ−ηa+e11−pξ=0.
Denotehξ=g′ξ, we have
(39)hξ=g′ξ=Aeαξ+Beξ1−p+D,where A=h0−2b2/αη2aη−θ+2b2/η2a21−p/1−α+αp. In view of F′′x<0, the value of A should be nonnegative. Then, hξ>0. If A is strictly positive, from (37) and (38), we get Fx~K0xα−1/α, where K0 is some fixed constant. Note that
(40)Fx≤E∫0∞er−ms−δBsux+θ+1as+bBs+λds≤Kxp+λe−rm−δ2/2,where K>0 is some fixed constant. Therefore, by Assumption 2, we must choose A to be equal zero. Recall that hξ=g′ξ and (37), we obtain the representation of the function
(41)Fx=λm−δ2/2+e−ξm−δ2/2−η−θa+η2a22b2g′ξ+1−ppeξ1−p=λm−δ2/2+e−ξm−δ2/2−η−θa+η2a22b2Beξ1−p+η2a22b2D+1−ppeξ1−p.
Thus, we havevr,x given in (30). Since F0=0 and g′ξ0=Beξ0/1−p+D, we get (31) and (32).
Since(42)qx=1−ηab2g′ξ,g′ξ>0,we get qx<1. Since g′′ξ=B/1−peξ/1−p>0, to satisfy the required condition qx>0, we need to show that the inequality ηa/b2g′ξ<1 is true. It is easy to verify that ξ∗ given in (34) is the unique solution of the equation ηa/b2g′ξ=1. In view of Assumption 2 and ηa/b2g′′ξ=2η−θ/αη1−p, we have ηa/b2g′ξ0<1, so we have 0<g′ξ<b2/ηa for ξ0<ξ<ξ∗, i.e., 0<qx<1 on 0,x∗.
It follows from the factsF′x=e−ξ>0 and F′′x=−e−ξ/g′ξ<0 that Fx is increasing and strictly concave. Substituting the values of F′ and F′′ into the left of (17), we have
(43)max0≤q≤1,c≥0θ−ηqaF′x+12b21−q2F′′x+δ22−mFx−cF′x+uc+λ=max0≤q≤1,c≥0θ−ηqae−ξ+12b21−q2e−ξg′ξ+δ22−mFx−ce−ξ+uc+λ.
Sinceu· is an increasing and concave function and 0<g′ξ<b2/ηa for ξ0<ξ<ξ∗, therefore, the maximizers q∗ and c∗ are given by q∗x=1−ηa/b2g′ξ and cx=eξ/1−p with x=gξ. Hence, Equation (43) can be rewritten as
(44)θ−ηae−ξ+η2a22b2g′ξe−ξ+δ22−mFx+1−ppep1−pξ+λ=0.
It follows from (30) and (31) that (44) is true. This finishes the proof.Sinceq∗x≤0 on the interval x∗,∞, it is reasonable to let q∗x=0; then, the HJB equation is equivalent to the following equation:
(45)θavx+12b2vxx+mvr+δ22vrr−cvx+e−ruc+λ=0.We also conjecture that the value function can be written in the formVx,r=e−rFx. In this case, the HJB equation of (45) becomes
(46)θaF′x+12b2F′′x−m−δ22Fx+1−ppF′x−p1−p+λ=0.with the boundary conditions
(47)F′x∗=e−ξ∗,(48)F′x∗F′′x∗=−b2ηa.If we setFx=Gx+λ/m−δ2/2, then the HJB equation of (46) becomes
(49)θaG′x+12b2G′′x−m−δ22Gx+1−ppG′x−p1−p=0,with the boundary conditions
(50)G′x∗=e−ξ∗andG′x∗G′′x∗=−b2ηa.Similar to the proof of Proposition 5 in Liang and Palmowski [17], we have the following results whose proof is omitted.Summarizing Propositions 4 and 5 gives the following result.Proposition 5.
Assume that Assumption2 holds. Suppose also that F on x∗,∞ solves (46)–(48). Then F′x>0, F′′x<0 on x∗,∞, and the the maximum in
(51)max0≤q≤1θ−ηqaF′x+12b21−q2F′′x=0is attained when q=0 in this case.Theorem 6.
Assume that Assumption2 holds. Then the value function Vr,x is given by
(52)Vr,x=e−rm−δ2/21α−1η−θa+1−p21−αp1−α+αpeξ1−pe−ξ+λ,if0<x≤x∗,e−rFx,ifx>x∗,with B and D given in (19) and (20) and x=gξ, where g is defined in (31). Moreover, x∗ is given in (33), and Fx solves (46). The corresponding optimal dividend strategy is
(53)c∗x=ervx−11−p,and the optimal reinsurance proportion is
(54)q∗x=1−ηab2g′ξ,if0<x≤x∗,0,ifx>x∗.Proof.
If0<x≤x∗, denote Vr,x=e−rFx, where
(55)Fx=λm−δ2/2+e−ξm−δ2/2−η−θa+η2a22b2Beξ1−p+η2a22b2D+1−ppeξ1−p.
SinceFx is a continuously differentiable solution of Equation (17), it is easy to obtain that e−rFx is a solution of (15). If x>x∗, denote Vr,x=e−rFx, where Fx is a continuously differentiable solution of Equation (36); it is easy to obtain that e−rFx is a solution of (15).
From now on, we prove the optimality of strategy(56)π∗=qt∗,ct∗=qXtπ∗,cXtπ∗.
Letπ be an admissible strategy with dividend and reinsurance strategy. Select 0<ξ1<x<ξ2. Define
(57)τx=inft≥0:Xtπ=x.
It follows from the Itô’s formula that we have(58)e−r−mt∧τξ1∧τξ2−δBt∧τξ1∧τξ2FXt∧τξ1∧τξ2π=∫0t∧τξ1∧τξ2e−r−ms−δBsF′Xsπθ−ηqsa+12b21−qs2F′′Xsπ−csF′Xsπds−∫0t∧τξ1∧τξ2e−r−ms−δBsm−δ22FXsπd+δ∫0t∧τξ1∧τξ2e−r−ms−δBsFXsπdBs+∫0t∧τξ1∧τξ2e−r−ms−δBsF′Xsπb−qsdWs≤e−rFx+∫0t∧τξ1∧τξ2e−r−ms−δBsF′Xsπb1−qsdWs+δ∫0t∧τξ1∧τξ2e−r−ms−δBsFXsπdBs−∫0t∧τξ1∧τξ2e−r−ms−δBsucs+λds.
Taking expectations on the both sides of the above inequality yields(59)Exe−r−mt∧τξ1∧τξ2−δBt∧τξ1∧τξ2FXt∧τξ1∧τξ2π≤e−rFx−Ex∫0t∧τξ1∧τξ2e−r−ms−δBsucs+λds.
Note that theπ=π∗ equality holds. We claim that e−r−mt−δBtFXtπt≥0 is uniformly integrable. Then, letting ξ1↓0, ξ2→∞, and by applying the dominated convergence theorem and the monotone convergence theorem, we have
(60)Ex∫0τe−r−ms−δBsucs∗+λds=e−rFx.
Thus,(61)e−rFx≥Vπr,x,ifπarbitrary,=Vπ∗r,x,ifπ=π∗.
And, we can conclude thatVr,x≥Vπ∗r,x=e−rFx≥supπVπr,x=Vr,x. By the concavity of F, we have
(62)Fx≤F′0x.
Thus, for eachT≥0,
(63)Exe−r−mT−δBTFXTπ∗≤Exe−r−mT−δBTsup0≤t≤TFXtπ∗≤Exe−r−mT−δBTF′0sup0≤t≤TFXtπ∗≤e−r−m−δ22TF′0Exsup0≤t≤TXtπ∗≤F′0Exsup0≤t≤TXtπ∗.
We introduce now a new stochastic processYt having the following dynamics:
(64)dYt=θadt+b1−qt∗dWt,Y0=x.
Then, one can observe thatYtt≥0 is a submartingale and Xtπ∗≤Yt for any t≥0. We denote YT∗=sup0≤t≤TYt. Then, by the Cauchy-Swartz inequality and Doob’s maximal inequality for submartingale, we have
(65)Exsup0≤t≤TXtπ∗≤ExYT∗≤ExYT∗2≤2ExYT2≤K1T+K2T<∞,where K1 and K2 are some fixed constants. Therefore, in view of (64) and (65), we can conclude that e−r−mt−δBtFXt∗ is uniformly integrable. This completes our proof.Remark 7.
Lettingm=β and r=δ=0 and λ→0 in Theorem 6, the result reduces to the result of Theorem 7 in Liang and Palmowski [17].


## 4. Solution to the Optimization Problem under VaR Constraints

In this section, we will use the results in Section3 to solve the optimal problem with VaR constraints (14). To do so, we make the following assumption.Assumption 8.
We assume thatVaR¯≥−Φ−1kbh−θah and Φ−1k<0 or equivalently k<1/2.Under Assumption8, the VaR constraint VaRtk,h,π≤VaR¯ is equivalent to
(66)cxh+Φ−1kbh+ηahqx−Φ−1kbh−θah−VaR¯≤0.Note that−Φ−1kbh−θah−VaR¯≤0, so there exists at least one strategy qx,cx≡0,0 that satisfies (66). Then, the control space defined in Assumption 8 is not empty. The strategy under VaR constraints should satisfy
(67)0≤qx≤1,cx≥0,cxh+Φ−1kbh+ηahqx−Φ−1kbh−θah−VaR¯≤0.If the strategyqx,cx defined in Theorem 6 satisfies the last inequality in (67), then the strategy q∗x,c∗x is also a solution with the constraint, namely q∗x,c∗x=qVaR∗x,cVaR∗x. Otherwise, if the strategy qx,cx defined in the first and second equations in (67) is outside the control space at the initial point x, then the optimal strategy is just the boundary of the control space. If the strategy qx,cx defined in the first and second equations in (67) is inside the control space at the initial point x, but it leaves the control space at sometime, we define the first exit point x¯ of the control space as
(68)x¯=infx>0:cxh+Eqx−F>0,orqx<0,orcx<0,where E=Φ−1kbh+ηah and F=VaR¯+Φ−1kbh+θah. Then, under the VaR constraints, the optimal strategy is πVaR∗=qVaR∗x,cVaR∗x with
(69)qVaR∗x=q∗x,ifx∈0,x¯∧x∗,q∗x¯∧x∗,ifx>x¯∧x∗,(70)cVaR∗x=c∗x,ifx∈0,x¯,c∗x¯,ifx>x¯.


## 5. Numerical Examples

In this section, we illustrate the results obtained in Sections3 and 4 by numerical examples. We set the model parameters of the insurers reserve process and the financial market in Table 1.Table 1
Parameters of the insurance market and the geometric Brownian motion.
abpηθrmδλ0.030.50.010.80.41210.5First, we plot the optimal dividend and optimal reinsurance proportion for surplusx ranging from 0 to 30 without VaR constraint. The strategies are shown in Figure 1. From Figure 1, we find the optimal threshold x∗ equals to 10.3337 according to Theorem 6, which implies that the insurer will take all the claims without buying any reinsurance when the surplus is larger than 10.3337. From this graph, we can also observe that the optimal dividend rate c∗x is nearly linear increasing with the surplus x, while the optimal reinsurance proportion is decreasing with x. When its surplus is sufficiently small, the insurer will divert most of its risk incurred by claims to the reinsurer, which seems intuitively reasonable. It is a reasonable result because when the insurer has a bigger initial wealth x, the insurer would like to retain a bigger proportion 1−qx of its insurance risks and increase the dividend rate.Figure 1
q∗x and c∗x change with x without VaR constraint.Second, we plot the optimal dividend and optimal reinsurance proportion as the functions ofp∈0,1 for fixed x=5 without VaR constraint. The strategies are shown in Figure 2. From Figure 2, we find that the optimal dividend rate increases with increasing p, while the optimal reinsurance proportion decreases. This phenomenon can be explained by noting that the parameter 1−p represents the risk aversion of the insurer, so the insurer will retain more proportion of the insurance risks and give more dividend payment.Figure 2
q∗x and c∗x change with p without VaR constraint.Third, we use the model parameters of the insurers reserve process market as in Table2 and consider the VaR constraints for three different cases as in Table 2, where the time interval h is equal to 1/250, and the VaR control levels of VaR¯ are set so that (66) holds. Cases 1 of Table 2: for the VaR control level VaR¯ and risk level k given in Case 1 of Table 2, time interval h=1/250, and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:
(71)qVaR∗x=q∗x,ifx∈0,7.1249,0.3085,ifx>7.1249,cVaR∗x=c∗x,ifx∈0,7.1249,7.2076,ifx>7.1249.Table 2
VaR control levels and levels with time intervalh=1/250.
Case 1Case 2Case 3VaR¯0.080.10.1k0.010.010.03Cases 2 of Table2: for the VaR control level VaR¯ and risk level k given in Case 2 of Table 2, time interval h=1/250, and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:
(72)qVaR∗x=q∗x,ifx∈0,8.9208,0.1353,ifx>8.9208,cVaR∗x=c∗x,ifx∈0,8.9208,9.0151,ifx>8.9208.Cases 3 of Table2: for the VaR control level VaR¯ and risk level k given in Case 3 of Table 2, time interval h=1/250, and the model parameter values given in Table 1, by using (69) and (70), we obtain the optimal reinsurance proportion and the optimal dividend strategy as follows:
(73)qVaR∗x=q∗x,ifx∈0,10.2729,0.0141,ifx>10.2729,cVaR∗x=c∗x,ifx∈0,10.2729,10.2791,ifx>10.2729.Under the VaR constraints, both the reinsurance strategy1−qVaR∗x and the cVaR∗x investment strategy are increasing while the initial capital x increases, which are presented in Figures 3 and 4. It is a reasonable result because when the insurer has a bigger initial wealth x, the insurer would like to retain a bigger proportion of its insurance risks and a larger dividend rate. The upper bounds represent the effect of the VaR constraints on the strategies, which implies that to limit the loss of the insurer at the VaR control level VaR¯, the insurer has to limit both the amounts of dividends and the retained insurance risks within certain scope. And, we can see that the upper bounds of the reinsurance strategy 1−qVaR∗x and the rate of dividend in case 2 are larger than those in case 1, which means that the insurer can retain more insurance risks and pay more dividend payment in Case 2 than in Case 1. This finding is consistent with the fact that the insurer has a tougher VaR control level of 0.08 in case 1 than that of the VaR control level 0.1 in case 2.Figure 3
q∗x change with x under the VaR constraints.Figure 4
c∗x change with x under the VaR constraints.



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*Source: 4051969-2020-08-11.xml*