# Solution of Hamilton-Jacobi-Bellman Equation in Optimal Reinsurance Strategy under Dynamic VaR Constraint

Authors: Yuzhen Wen; Chuancun Yin
Journal: Journal of Function Spaces
 (2019)

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## Abstract

This paper analyzes the optimal reinsurance strategy for insurers with a generalized mean-variance premium principle. The surplus process of the insurer is described by the diffusion model which is an approximation of the classical Cramér-Lunderberg model. We assume the dynamic VaR constraints for proportional reinsurance. We obtain the closed form expression of the optimal reinsurance strategy and corresponding survival probability under proportional reinsurance.


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## Body

## 1. Introduction

In practice, reinsurance is an important way for an insurer to control its risk exposure. In the actuarial literature, the optimal reinsurance problem of minimising ruin probability or equivalently maximising survival probability has been studied extensively in the past two decades. As one type of typical reinsurance strategy, proportional reinsurance has received great attention from both the academics and practitioners. Among others, Choulli et al. (2003), Højgaard and Taksar [1, 2], Schmidli [3, 4], Taksar [5], and Zhang et al. [6] work on the proportional reinsurance.In the existing literature, the expected value principle is commonly used as the reinsurance premium principle due to its simplicity and popularity in practice. For details, the readers are referred to Bäuerle [7], Bai and Zhang [8], and Liang and Bayraktar [9]. Generally speaking, expected value principle is commonly used in life insurance whose claim frequency and claim sizes are stable and smooth, while the variance premium principle is extensively used in property insurance; see Zhou and Yuen [10] and Sun et al. [11]. Similarly to Zhang et al. [6], in this paper, we focus on a generalized mean-variance premium principle, which includes the expected value principle and the variance principle as special cases.More recently, the problem of optimal reinsurance design has been studied by using risk measures such as the Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and conditional tail expectation (CTE) (to name a few, Cai and Tan [12], Cheung et al. [13], and Cai et al. [14, 15]). Latterly static risk measures have been extended to the dynamic version; see Yiu [16], Alexander and Baptista [17], Cuoco et al. [18], Chen et al. [19], and Zhang et al. [6], all of which investigate the optimal reinsurance problem under dynamic VaR constraint.In this paper, we investigate an optimal proportional reinsurance problem under dynamic VaR constraint. Assume that an insurer aims to maximize the survival probability. With this assumption, we obtain the closed form expressions. The rest of the paper is organized as follows. In Section2, we provide a general formulation of the optimal reinsurance problem. Then we investigate the insurance company’s maximum survival probability under dynamic VaR constraints, and the corresponding optimal reinsurance strategy is given in proportional reinsurance settings in Section 3.


## 2. Formulation

Let(Ω,F,P) be a probability space with a filtration {Ft}t≥0. Consider a Cramér-Lundberg model with the surplus process of an insurance company being given by (1)Ut=u0+ct-∑i=1NtYi=u0+ct-St,where u0 is the initial surplus, the claim arrival process {N(t)}t≥0 is a Poisson process with constant intensity λ>0, and the random variables Yi, i=1,2,…, are i.i.d claim sizes independent of N(t). We let Ti denote the i-th claim occurrence time and F(y) denote the claim size distribution with finite first and second moments m1, m2. The premium rate c is assumed to be calculated via the expected value principle; that is, (2)c=1+ηλm1,where η>0 is the relative loading factor.In this paper, the insurer can purchase proportional reinsurance to adjust the exposure to insurance risk. The proportional reinsurance level is associated with the risk exposureq(t) at time t. We assume q(t)∈[0,1] for all t, and it means the insurer purchases proportional reinsurance. In this case, for each claim, the insurer only pays its q(t)Y, while the reinsurer pays the rest (1-q(t))Y for each claim.For a chosen reinsurance policyq(t), let {Uq(t),t≥0} denote the associated surplus process; that is, Uq(t) is the surplus of insurer at time t. This process then evolves as (3)dUqt=c-cqdt-qtd∑i=1NtYi,where cq is the net reinsurance rate which the reinsurer receives from the insurer. We assume that the reinsurance premium is calculated by the following generalized mean-variance principle (1+θ)[E(·)+ζD(·)], where θ,ζ≥0, and E and D denote the expectation and variance, respectively. Thus we have(4)cqt=1+θλ1-qtm1+ζ1-qt2m2t,and the premium rate for the insurer is(5)c-cq=η-θλm1+1+θλqtm1+λ1+θζ1-qt2m2.According to Grandell (1991), the surplus process after reinsurance can be approximated by the following diffusion process:(6)Uqt=u+∫0tη-θλm1+θλqsm1+λ1+θζ1-qs2m2ds+∫0tqsλm2dBs,where {B(t)}t≥0 is a standard Brownian motion.We define the ruin time(7)τq=inft>0,Uqt≤0where the superscript q emphasises that the surplus process and the ruin time are controlled by an admissible policy q. Denote the survival probability given the initial surplus u by (8)υqu=Pτq=∞∣Uq0=u,and the maximum survival probability by (9)υu=maxq∈Πυqu.Our objective is to find the value functionυ(u) and the optimal policy q∗ such that (10)υu=υq∗u.


## 3. Maximizing Survival Probability

Under the proportional reinsurance, the insurer could transfer a fraction1-q(t) of the incoming claims to a reinsurer, where q(t) is Ft-measurable and satisfies 0≤q(t)≤1 for all t. The diffusion approximation of insurance company’s claim process becomes (11)dSqt=λqtm1dt+qtλm2dBt,Sq0=0,where B(t) is a standard Brownian motion. The insurer’s surplus process satisfies the stochastic differential equation (12)dUqt=η-θλm1+θλqsm1+λ1+θζ1-qs2m2dt+qtλm2dBt,Uq0=u0.Takingh>0 is small enough, we assume that risk exposure does not change over the short time period [t,t+h]. This means that the risk exposure remains roughly constant in the given time period; that is, E[q(s)Y]=E[q(t)Y], λE[(q(s)Y)2]=λE[(q(t)Y)2], s∈[t,t+h]. This setting is reasonable because the insurer can only adjust its reinsurance business at discrete time; and the decision made is based on the holding at time t. Thus, we rewrite the claim dynamics as(13)Sqt+h-Sqt=∫tt+hλEqsYds+∫tt+hλEqsY2dBs≔λqtm1h+qtλm2∫tt+hdBs.


### 3.1. Dynamic VaR, CVaR, and Worst-Case CVaR

For a given confidence level1-α∈(0,1) and a given horizon h>0, the VaR at time t of a proportional reinsurance policy q, denoted by VaRtα,h, is defined as (14)VaRtα,h≜inf⁡L:PSqt+h-Sqt≥L∣Ft<α.The dynamic Conditional Value-at-Risk CVaRtα,h is given by (15)CVaRtα,h≜ESqt+h-Sqt∣Sqt+h-Sqt≥VaRtα,h.The dynamic worst-case CVaR is defined as (16)wcCVaRtα,h≜supp·∈P1infa∈Ra+1αEpSqt+h-Sqt-a+,where(17)P1=p·:EpSqt+h-Sqt=λhm1qt,EpSqt+h-Sqt2=λhm2qt2+λhm1qt2Proposition 1 (Zhang et al. [6]).
(18) V a R t α , h = λ h q t m 1 - Φ - 1 α q t λ h m 2 , C V a R t α , h = λ h q t m 1 + ϕ Φ - 1 α α q t λ h m 2 , w c C V a R t α , h = λ h q t m 1 + 1 - α α q t λ h m 2 , 0 ≤ V a R t α , h ≤ C V a R t α , h ≤ w c C V a R t α , h < U q t ,where ϕ(x) and Φ(x) denote the probability density function and the cumulative distribution function of a standard normal random variable, respectively. Φ-1(x) is the inverse function of Φ(x).


### 3.2. HJB Equation

Using the dynamic programming technique, we obtain that the value functionυ(u) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:(19)maxq∈0,1η-θλm1+θλqm1-λ1+θζ1-q2m2υ′u+12λq2m2υ′′u=0,(20)s.t.q∈0,1;VaRtα,h≤ku;υ0=0,υ+∞=1,where k (0<k<+∞) is a constant.Next we try to construct a solution of the HJB equation (19) with the boundary condition (20). Suppose that ηm1≤θm1+(1+θ)ζm2, υ(u) with υ′(u)>0, υ′′(u)≠0 satisfies (19) and (20).Theorem 2.
(a) Ifθ≥2η, the function(21)φu=φAk-φAk∫uA/kw-△2e-△1/we△3wdw∫0A/kw-△2e-△1/we△3wdw,if0<u≤Ak,φAk+1-φAk1-e-2ηm1/m2u-A/k,ifu≥Ak,is a smooth (C2) solution to the HJB equation, where(22)A=λm1h-Φ-1αλm2h,φAk=2ηm1/m22ηm1/m2+A/k-△2e-k△1/AeA△3/k/∫0A/kw-△2e-△1/we△3wdw,△1=2A2[(η-θ)m1-(1+θ)ζm2]/k2m2, △2=2A[θm1+2(1+θ)ζm2]/km2, △3=2(1+θ)ζ. The maximum of the left side of HJB equation is attained at(23)q∗u=kuA,ifu<Ak,1,ifu≥Ak.(b) If θ<2η, the function(24)φu=φu1-φu1∫uu1w-△2e-△1/we△3wdw∫0u1w-△2e-△1/we△3wdw,if0<u≤u1,φu1+1-φu11-e-△4u-u1,ifu≥u1,is a smooth (C2) solution to the HJB equation, where(25)φu1=△4△4+u1-△2e-△1/u1e△3u1/∫0u1w-△2e-△1/we△3wdw,△4=1/2θ2m12+2ζη(1+θ)m1m2/m2[ζ(1+θ)m2+(θ-η)m1], u1=A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1. The maximum of the left side of HJB equation is attained at (26)q∗u=kuA,ifu<u1,θ-ηm1+1+θζm21+θζm2+1/2θm1,ifu≥u1.Proof.
We solve the HJB equation analytically. First we need to determine the optimal strategyq∗(u). Differentiating the terms inside the maximum in (19) with respect to q(u) and setting to 0 yield(27)q0u=θm1+2ζ1+θm22ζ1+θm2-m2φ′′/φ′.
The dynamic VaR constraint impliesq(u)≤ku/A, when A is defined by (27). Normally, we take 0<α<1/2; hence, A is always positive.
(1) Foru≥A/k, we have ku/A≥1. Then, from q(u)≤ku/A obtained from the dynamic VaR constraint and the requirement that the retained proportion of claims q(u) is always within [0,1], we have q(u)∈[0,1].
(a) Ifq0(u)≥1, we let q∗(u)=1, and then the HJB equation becomes(28)ηm1φ′u+12m2φ′′u=0,which implies(29)φ′′uφ′u=-2ηm1m2.Inserting it into (28), we obtain(30)q0u=θm1+2ζ1+θm22ζ1+θm2+2ηm1.(i)
Ifθ≥2η, we have q0(u)≥1; consequently q∗(u)=1, and then the HJB equation becomes (30).(ii)
Ifθ<2η, we have q0(u)<1, where conflict exits.
(b) Ifq0(u)<1, we have q∗(u)=q0(u), and then the HJB equation becomes(31)φ′′uφ′u=-1/2θ2m12+2ζη1+θm1m2m2ζ1+θm2+θ-ηm1,which implies(32)q0u=ζ1+θm2+θ-ηm11/2θm1+ζ1+θm2.(i)
Ifθ≥2η, we have q0(u)≥1, where conflict exits.(ii)
Ifθ<2η, we have q0(u)<1; consequently q∗(u)=q0(u)=ζ(1+θ)m2+(θ-η)m1/1/2θm1+ζ(1+θ)m2, and then the HJB equation becomes (32).
(2) When0<u<A/k, we have ku/A<1; thus 0≤q(u)≤ku/A.
(a) Ifq0(u)≥ku/A, we have q∗(u)=ku/A, and then the HJB equation becomes (33)η-θm1+θkuAm1-ζ1+θ1-kuA2m2φ′u+12m2kuA2φ′′u=0.We have(34)φ′′uφ′u=-η-θm1+θku/Am1-ζ1+θ1-ku/A2m21/2m2ku/A2,which implies(35)q0u=12kuA21ku/A-θ-ηm1+1+θζm2/θm1+2ζ1+θm2.
We haveφ′′(u)<0 when u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), and we have q0(u)>0 when u>A/k(θ-η)m1+(1+θ)ζm2/2(1+θ)ζm2+θm1. We have φ′′(u)<0 and q0(u)>0 when u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1+θ2m12+41+θζηm1m2.
ForA/k>u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), we have the following. (i)
Ifθ≥2η, we have q0(u)≥ku/A; consequently q∗(u)=q0(u)=ku/A, and then the HJB equation becomes (34).(ii)
Ifθ<2η, we have q0(u)<ku/A, where conflict exits.
Foru≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), we have φ′′(u)≥0; therefore φ(u) is convex for small u. Through the analysis of the HJB equation (19), for 0≤u≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), the maximum of the left side of the HJB is attained at q∗(u)=ku/A and the HJB equation becomes (34).
(b) Whenq0(u)≤ku/A, it is reasonable to let q∗(u)=q0(u). Similar to (a), we have the following conclusions.
For0≤u≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, the optimal strategy is obtained at q∗(u)=ku/A.
ForA/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1<u≤A/k, we have the following. (i)
Ifθ≥2η, we have q0(u)≥1>ku/A, where conflict exits.(ii)
Ifθ<2η, we have q0(u)<ku/A; consequently, q∗(u)=q0(u)=(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, and then the HJB equation becomes (32).
From the previous analysis, we have the following conclusions.(i)
Ifθ≥2η, the maximum of the left side of HJB equation is attained at(36)q∗u=kuA,ifu<Ak,1,ifu≥Ak.(ii)
Ifθ<2η, the maximum of the left side of HJB equation is attained at(37)q∗u=kuA,ifu<u1,θ-ηm1+1+θζm21+θζm2+1/2θm1,ifu≥u1.In the following, we will solve the HJB equation in each situation.
Forθ≥2η and u<A/k, the HJB equation is (34), which is equivalent to (35). Taking integral from u to A/k, we obtain(38)φu=φAk-K∫uA/kw-△2e-△1/we△3wdw,where △1=2A2[(η-θ)m1-(1+θ)ζm2]/k2m2, △2=2A[θm1+2(1+θ)ζm2]/km2, and △3=2(1+θ)ζ. Applying the boundary condition φ(0)=0 we obtain(39)K=φA/k∫0A/kw-△2e-△1/we△3wdw.
Forθ≥2η and u≥A/k, the corresponding HJB is (29), which is equivalent to (30). Taking integral form A/k to u, we obtain(40)φu=φAk+m22ηm1φ′Ak1-e-2ηm1/m2u-A/k.Applying the boundary condition δ(+∞)=1 we obtain(41)φu=φAk+1-φA/k1-e-2ηm1/m2u-A/k.Considering that φ(x) is twice continuously differentiable, it should satisfy φ′(A/k-)=φ′(A/k+); that is,(42)φA/kA/k-△2e-k△1/AeA△3/k∫0A/kw-△2e-△1/we△3wdw=2ηm1m21-φAk,which leads to(43)φAk=2ηm1/m22ηm1/m2+A/k-△2e-k△1/AeA△3/k/∫0A/kw-△2e-△1/we△3wdw.Thus, if θ≥2η, we have the function(44)φu=φAk-φAk∫uA/kw-△2e-△1/we△3wdw∫0A/kw-△2e-△1/we△3wdw,if0<u≤Ak,φAk+1-φAk1-e-2ηm1/m2u-A/k,ifu≥Ak.
Ifθ<2η and u<u1, the HJB equation is (34), and the HJB equation is (32) for θ<2η and u≥u1. From the procedure that is similar to the previous analysis we can get the following function is a C2 solution to HJB; that is,(45)φu=φu1-φu1∫uu1w-△2e-△1/we△3wdw∫0u1w-△2e-△1/we△3wdw,if0<u≤u1,φu1+1-φu11-e-△4u-u1,ifu≥u1,where(46)△4=1/2θ2m12+2ζη1+θm1m2m2ζ1+θm2+θ-ηm1,φu1=△4△4+u1-△2e-△1/u1e△3u1/∫0u1w-△2e-△1/we△3wdw.This ends the proof.When the value function is twice continuously differentiable, then it is the unique solution of the HJB equation (see, e.g., [20]), and we have the following result.Proposition 3.
The value functionv(x) coincides with the smooth function φ(x) defined in Theorem 2 and the optimal control, which represents the optimal proportional reinsurance strategy, is described by the q∗(u) in Theorem 2, where {Uq∗(t),t≥0} is the corresponding surplus process.Remark 4.
Whenζ=0, Theorem 2 coincides with theorem 3.1 in Zhang et al. [6].Corollary 5.
Whenθ=0, the generalized mean-variance premium principle is mean-variance premium principle, and we have the following:(47)φu=φu1-φu1∫uu1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw∫0u1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw,if0<u≤u1,φu1+1-φu11-e-2ζηm1/ζm2-ηm1u-u1,ifu≥u1,is a smooth (C2) solution to the HJB equation, where(48)φu1=2ζηm1/ζm2-ηm12ζηm1/ζm2-ηm1+u1-4Aζ/ke-2A2ηm1-ζm2/u1k2m2e2ζu1/∫0u1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw,u1=A/k(1-ηm1/ζm2). The maximum of the left side of HJB equation is attained at (49)q∗u=kuA,ifu<u1,1-ηm1ζm2,ifu≥u1.Corollary 6.
When there is no dynamic VaR, CVaR, or wcCVaR constraints, that is,k=∞, and the model becomes the unconstrained reinsurance problem, we have the following.
(a) Ifθ≥2η, the optimal reinsurance strategy is q∗=1, and the optimal survival probability is(50)φu=1-e-2ηm1/m2u.(b) If θ<2η, the optimal reinsurance strategy is q∗=(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, and the optimal survival probability is (51)φu=1-e-1/2θ2m12+2ζη1+θm1m2/m2ζ1+θm2+θ-ηm1u.


## 3.1. Dynamic VaR, CVaR, and Worst-Case CVaR

For a given confidence level1-α∈(0,1) and a given horizon h>0, the VaR at time t of a proportional reinsurance policy q, denoted by VaRtα,h, is defined as (14)VaRtα,h≜inf⁡L:PSqt+h-Sqt≥L∣Ft<α.The dynamic Conditional Value-at-Risk CVaRtα,h is given by (15)CVaRtα,h≜ESqt+h-Sqt∣Sqt+h-Sqt≥VaRtα,h.The dynamic worst-case CVaR is defined as (16)wcCVaRtα,h≜supp·∈P1infa∈Ra+1αEpSqt+h-Sqt-a+,where(17)P1=p·:EpSqt+h-Sqt=λhm1qt,EpSqt+h-Sqt2=λhm2qt2+λhm1qt2Proposition 1 (Zhang et al. [6]).
(18) V a R t α , h = λ h q t m 1 - Φ - 1 α q t λ h m 2 , C V a R t α , h = λ h q t m 1 + ϕ Φ - 1 α α q t λ h m 2 , w c C V a R t α , h = λ h q t m 1 + 1 - α α q t λ h m 2 , 0 ≤ V a R t α , h ≤ C V a R t α , h ≤ w c C V a R t α , h < U q t ,where ϕ(x) and Φ(x) denote the probability density function and the cumulative distribution function of a standard normal random variable, respectively. Φ-1(x) is the inverse function of Φ(x).


## 3.2. HJB Equation

Using the dynamic programming technique, we obtain that the value functionυ(u) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:(19)maxq∈0,1η-θλm1+θλqm1-λ1+θζ1-q2m2υ′u+12λq2m2υ′′u=0,(20)s.t.q∈0,1;VaRtα,h≤ku;υ0=0,υ+∞=1,where k (0<k<+∞) is a constant.Next we try to construct a solution of the HJB equation (19) with the boundary condition (20). Suppose that ηm1≤θm1+(1+θ)ζm2, υ(u) with υ′(u)>0, υ′′(u)≠0 satisfies (19) and (20).Theorem 2.
(a) Ifθ≥2η, the function(21)φu=φAk-φAk∫uA/kw-△2e-△1/we△3wdw∫0A/kw-△2e-△1/we△3wdw,if0<u≤Ak,φAk+1-φAk1-e-2ηm1/m2u-A/k,ifu≥Ak,is a smooth (C2) solution to the HJB equation, where(22)A=λm1h-Φ-1αλm2h,φAk=2ηm1/m22ηm1/m2+A/k-△2e-k△1/AeA△3/k/∫0A/kw-△2e-△1/we△3wdw,△1=2A2[(η-θ)m1-(1+θ)ζm2]/k2m2, △2=2A[θm1+2(1+θ)ζm2]/km2, △3=2(1+θ)ζ. The maximum of the left side of HJB equation is attained at(23)q∗u=kuA,ifu<Ak,1,ifu≥Ak.(b) If θ<2η, the function(24)φu=φu1-φu1∫uu1w-△2e-△1/we△3wdw∫0u1w-△2e-△1/we△3wdw,if0<u≤u1,φu1+1-φu11-e-△4u-u1,ifu≥u1,is a smooth (C2) solution to the HJB equation, where(25)φu1=△4△4+u1-△2e-△1/u1e△3u1/∫0u1w-△2e-△1/we△3wdw,△4=1/2θ2m12+2ζη(1+θ)m1m2/m2[ζ(1+θ)m2+(θ-η)m1], u1=A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1. The maximum of the left side of HJB equation is attained at (26)q∗u=kuA,ifu<u1,θ-ηm1+1+θζm21+θζm2+1/2θm1,ifu≥u1.Proof.
We solve the HJB equation analytically. First we need to determine the optimal strategyq∗(u). Differentiating the terms inside the maximum in (19) with respect to q(u) and setting to 0 yield(27)q0u=θm1+2ζ1+θm22ζ1+θm2-m2φ′′/φ′.
The dynamic VaR constraint impliesq(u)≤ku/A, when A is defined by (27). Normally, we take 0<α<1/2; hence, A is always positive.
(1) Foru≥A/k, we have ku/A≥1. Then, from q(u)≤ku/A obtained from the dynamic VaR constraint and the requirement that the retained proportion of claims q(u) is always within [0,1], we have q(u)∈[0,1].
(a) Ifq0(u)≥1, we let q∗(u)=1, and then the HJB equation becomes(28)ηm1φ′u+12m2φ′′u=0,which implies(29)φ′′uφ′u=-2ηm1m2.Inserting it into (28), we obtain(30)q0u=θm1+2ζ1+θm22ζ1+θm2+2ηm1.(i)
Ifθ≥2η, we have q0(u)≥1; consequently q∗(u)=1, and then the HJB equation becomes (30).(ii)
Ifθ<2η, we have q0(u)<1, where conflict exits.
(b) Ifq0(u)<1, we have q∗(u)=q0(u), and then the HJB equation becomes(31)φ′′uφ′u=-1/2θ2m12+2ζη1+θm1m2m2ζ1+θm2+θ-ηm1,which implies(32)q0u=ζ1+θm2+θ-ηm11/2θm1+ζ1+θm2.(i)
Ifθ≥2η, we have q0(u)≥1, where conflict exits.(ii)
Ifθ<2η, we have q0(u)<1; consequently q∗(u)=q0(u)=ζ(1+θ)m2+(θ-η)m1/1/2θm1+ζ(1+θ)m2, and then the HJB equation becomes (32).
(2) When0<u<A/k, we have ku/A<1; thus 0≤q(u)≤ku/A.
(a) Ifq0(u)≥ku/A, we have q∗(u)=ku/A, and then the HJB equation becomes (33)η-θm1+θkuAm1-ζ1+θ1-kuA2m2φ′u+12m2kuA2φ′′u=0.We have(34)φ′′uφ′u=-η-θm1+θku/Am1-ζ1+θ1-ku/A2m21/2m2ku/A2,which implies(35)q0u=12kuA21ku/A-θ-ηm1+1+θζm2/θm1+2ζ1+θm2.
We haveφ′′(u)<0 when u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), and we have q0(u)>0 when u>A/k(θ-η)m1+(1+θ)ζm2/2(1+θ)ζm2+θm1. We have φ′′(u)<0 and q0(u)>0 when u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1+θ2m12+41+θζηm1m2.
ForA/k>u>A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), we have the following. (i)
Ifθ≥2η, we have q0(u)≥ku/A; consequently q∗(u)=q0(u)=ku/A, and then the HJB equation becomes (34).(ii)
Ifθ<2η, we have q0(u)<ku/A, where conflict exits.
Foru≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), we have φ′′(u)≥0; therefore φ(u) is convex for small u. Through the analysis of the HJB equation (19), for 0≤u≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2(θm1+θ2m12+4(1+θ)ζηm1m2), the maximum of the left side of the HJB is attained at q∗(u)=ku/A and the HJB equation becomes (34).
(b) Whenq0(u)≤ku/A, it is reasonable to let q∗(u)=q0(u). Similar to (a), we have the following conclusions.
For0≤u≤A/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, the optimal strategy is obtained at q∗(u)=ku/A.
ForA/k(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1<u≤A/k, we have the following. (i)
Ifθ≥2η, we have q0(u)≥1>ku/A, where conflict exits.(ii)
Ifθ<2η, we have q0(u)<ku/A; consequently, q∗(u)=q0(u)=(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, and then the HJB equation becomes (32).
From the previous analysis, we have the following conclusions.(i)
Ifθ≥2η, the maximum of the left side of HJB equation is attained at(36)q∗u=kuA,ifu<Ak,1,ifu≥Ak.(ii)
Ifθ<2η, the maximum of the left side of HJB equation is attained at(37)q∗u=kuA,ifu<u1,θ-ηm1+1+θζm21+θζm2+1/2θm1,ifu≥u1.In the following, we will solve the HJB equation in each situation.
Forθ≥2η and u<A/k, the HJB equation is (34), which is equivalent to (35). Taking integral from u to A/k, we obtain(38)φu=φAk-K∫uA/kw-△2e-△1/we△3wdw,where △1=2A2[(η-θ)m1-(1+θ)ζm2]/k2m2, △2=2A[θm1+2(1+θ)ζm2]/km2, and △3=2(1+θ)ζ. Applying the boundary condition φ(0)=0 we obtain(39)K=φA/k∫0A/kw-△2e-△1/we△3wdw.
Forθ≥2η and u≥A/k, the corresponding HJB is (29), which is equivalent to (30). Taking integral form A/k to u, we obtain(40)φu=φAk+m22ηm1φ′Ak1-e-2ηm1/m2u-A/k.Applying the boundary condition δ(+∞)=1 we obtain(41)φu=φAk+1-φA/k1-e-2ηm1/m2u-A/k.Considering that φ(x) is twice continuously differentiable, it should satisfy φ′(A/k-)=φ′(A/k+); that is,(42)φA/kA/k-△2e-k△1/AeA△3/k∫0A/kw-△2e-△1/we△3wdw=2ηm1m21-φAk,which leads to(43)φAk=2ηm1/m22ηm1/m2+A/k-△2e-k△1/AeA△3/k/∫0A/kw-△2e-△1/we△3wdw.Thus, if θ≥2η, we have the function(44)φu=φAk-φAk∫uA/kw-△2e-△1/we△3wdw∫0A/kw-△2e-△1/we△3wdw,if0<u≤Ak,φAk+1-φAk1-e-2ηm1/m2u-A/k,ifu≥Ak.
Ifθ<2η and u<u1, the HJB equation is (34), and the HJB equation is (32) for θ<2η and u≥u1. From the procedure that is similar to the previous analysis we can get the following function is a C2 solution to HJB; that is,(45)φu=φu1-φu1∫uu1w-△2e-△1/we△3wdw∫0u1w-△2e-△1/we△3wdw,if0<u≤u1,φu1+1-φu11-e-△4u-u1,ifu≥u1,where(46)△4=1/2θ2m12+2ζη1+θm1m2m2ζ1+θm2+θ-ηm1,φu1=△4△4+u1-△2e-△1/u1e△3u1/∫0u1w-△2e-△1/we△3wdw.This ends the proof.When the value function is twice continuously differentiable, then it is the unique solution of the HJB equation (see, e.g., [20]), and we have the following result.Proposition 3.
The value functionv(x) coincides with the smooth function φ(x) defined in Theorem 2 and the optimal control, which represents the optimal proportional reinsurance strategy, is described by the q∗(u) in Theorem 2, where {Uq∗(t),t≥0} is the corresponding surplus process.Remark 4.
Whenζ=0, Theorem 2 coincides with theorem 3.1 in Zhang et al. [6].Corollary 5.
Whenθ=0, the generalized mean-variance premium principle is mean-variance premium principle, and we have the following:(47)φu=φu1-φu1∫uu1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw∫0u1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw,if0<u≤u1,φu1+1-φu11-e-2ζηm1/ζm2-ηm1u-u1,ifu≥u1,is a smooth (C2) solution to the HJB equation, where(48)φu1=2ζηm1/ζm2-ηm12ζηm1/ζm2-ηm1+u1-4Aζ/ke-2A2ηm1-ζm2/u1k2m2e2ζu1/∫0u1w-4Aζ/ke-2A2ηm1-ζm2/wk2m2e2ζwdw,u1=A/k(1-ηm1/ζm2). The maximum of the left side of HJB equation is attained at (49)q∗u=kuA,ifu<u1,1-ηm1ζm2,ifu≥u1.Corollary 6.
When there is no dynamic VaR, CVaR, or wcCVaR constraints, that is,k=∞, and the model becomes the unconstrained reinsurance problem, we have the following.
(a) Ifθ≥2η, the optimal reinsurance strategy is q∗=1, and the optimal survival probability is(50)φu=1-e-2ηm1/m2u.(b) If θ<2η, the optimal reinsurance strategy is q∗=(θ-η)m1+(1+θ)ζm2/(1+θ)ζm2+1/2θm1, and the optimal survival probability is (51)φu=1-e-1/2θ2m12+2ζη1+θm1m2/m2ζ1+θm2+θ-ηm1u.



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*Source: 6750892-2019-01-08.xml*