# Optimal Strategies for an Ambiguity-Averse Insurer under a Jump-Diffusion Model and Defaultable Risk

Authors: Man Li; Yingchun Deng; Ya Huang; Hui Ou
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, we consider a robust optimal investment-reinsurance problem with a default risk. The ambiguity-averse insurer (AAI) may carry out transactions on a risk-free asset, a stock, and a defaultable corporate bond. The stock’s price is described by a jump-diffusion process, and both the jump intensity and the distribution of jump amplitude are uncertain, i.e., the jump is ambiguous. The AAI’s surplus process is assumed to follow an approximate diffusion process. In particular, the reinsurance premium is calculated according to the generalized mean-variance premium principle, and the reinsurance type has to follow a self-reinsurance function. In performing dynamic programming, both the predefault case and the postdefault case are analyzed, and the optimal strategies and the corresponding value functions are derived under the worst-case scenario. Moreover, we give a detailed proof of the verification theorem and give some special cases and numerical examples to illustrate our theoretical results.


---

## Body

## 1. Introduction

Investment is the most common way for the insurer to cope with the fierce competition in the insurance market and get higher returns, including risk-free investment (bank), risky investment (stock), and bond investment. The insurer can also transfer their risks by buying reinsurance. Therefore, the optimal investment-reinsurance problem of insurers has received extensive attention in the field of insurance and stochastic control. Browne [1] originally optimized the exponential utility of terminal wealth in order to obtain an optimal investment strategy for the insurer. Since then, a large number of works have been done concerning this topic (see Yang and Zhang [2], Bai and Guo [3], Huang et al. [4], Zhang et al. [5], Zeng et al. [6], Peng et al. [7], etc.).For the stock’s price process, some scholars have paid attention to the jump risk, such as Yu et al. [8] and Zhang et al. [9]. This is because in the face of serious events (natural disasters and serious large-scale diseases), the stock price may jump to a new level. Therefore, it is not suitable for the stock price to be described by a geometric Brownian motion (GBM) with the constant appreciation rate and volatility. Moreover, the optimal investment-reinsurance problem under the jump-diffusion model has drawn much attention, for example, Li et al. [10], Cao [11], Liang et al. [12], Lin et al. [13], Yang and Zhang [2], and Zhao et al. [14].Recently, the investors/insurers have an increasing interest in the default risk of corporate bonds with high yield. Default risk (credit risk) refers to the risk that the security issuer will not be able to repay the principal and interest at the maturity of the security, which makes the investor suffer losses. Therefore, it is the purpose of investors to reduce credit risk and obtain higher returns. In fact, several scholars have addressed the portfolio optimization on corporate bonds in the last several decades. Bielecki and Jang [15] studied an optimal allocation problem associated with defaultable bond, and their goal was to maximize the expected utility of the terminal wealth. Bo et al. [16, 17] considered an investment-consumption problem for an investor who can invest in a defaultable market. For more results about default risk, see Capponi and Figueroa-López [18], Zhu et al. [19], Zhao et al. [20], and Deng et al. [21].For the insurer, reinsurance is an important method to balance their risk and obtain higher profit via an optimal reinsurance strategy. The reinsurance includes the reinsurance premium and reinsurance type. In most of the above results, the reinsurance premium principle is calculated according to the expected value principle or variance principle, such as Liang and Bayraktar [22] and Sun et al. [23]. In previous conclusions, two types of reinsurance policies are most commonly studied in the literature. One policy is the proportional (quota-share) reinsurance (see Zhou et al. [24] and Shen and Zeng [25]), and the other policy is the excess-of-loss reinsurance (see [10, 14]). Recently, Zhang et al. [26] studied the optimal investment-insurance for insurers with the generalized mean-variance principle. In their article, a generalized mean-variance principle included two special cases: the expected value principle and the variance principle. The reinsurance policy was considered a self-reinsurance function, which included the proportional reinsurance and the excess-of-loss reinsurance.Apart from the investment-reinsurance, recent advances are on the ambiguity aversion, the uncertainty associated with the model, and the risk aversion. In reality, it is a notorious fact that the return of risky assets is difficult to estimate accurately. Therefore, investors may consider some alternative models that are close to the estimated model to deal with portfolio selection in case of ambiguity. Anderson et al. [27] introduced the concept of ambiguity aversion and formulated a robust control problem for investors. Uppal and Wang [28] extended the results of Anderson et al. [27] under the model uncertainty robustness framework with different levels of ambiguity. For investors, Maenhout [29, 30] derived the closed-form solutions to robust optimal strategies by innovating a “homothetic robustness” framework. A lot of descendent researches of Maenhout [29, 30] concentrated on the influences of ambiguity in the field of finance or insurance, and the representative publications are Zhang and Siu [31], Yi et al. [32], Flor and Larsen [33], Pun and Wong [34], Zeng et al. [35], Zheng et al. [26], Zhang et al. [36], etc.Moreover, the jump risk, especially that associated with disaster events, is more difficult to estimate accurately. So many scholars have paid attention to the ambiguity of jump risks. For the sake of explanation, the distribution of jump amplitude is assumed to be known and is restricted to be identical under the reference model’s measureP and the alternative measure Pϕ, but the jump intensity is uncertain. This topic was studied by some scholars, for example, Branger and Larsen [37], Zeng et al. [6], Sun et al. [23], and Li et al. [10]. But in most instances, the distribution of jump amplitude is unknown. Jin et al. [38] considered the dynamic portfolio choice problem with ambiguous jump risks in a multidimensional jump-diffusion framework. In their results, both the jump amplitude distribution and the jump intensity were assumed to be uncertain.For these reasons, we choose a jump-diffusion process to describe the price of the stock and consider the robust model to find an optimal strategy in this paper. Moreover, the AAI is allowed to buy reinsurance and allocate his/her wealth among a risk-free asset, a stock, and a default corporate bond. According to Zhang et al. [5], we assume that the reinsurance premium is calculated about the generalized mean-variance principle, which is more general than that reported by Sun et al. [23] and Li et al. [10]. Specifically, in our model, the stock’s price changes dramatically, while the parameters of the underlying jump processes are difficult to estimate accurately. Therefore, we assume that the stock’s jump amplitude distribution and the jump intensity are uncertain, which are different from those reported by Sun et al. [23] and Li et al. [10]. The surplus of an AAI is described by an approximate diffusion process. In light of the principle of dynamic programming, the corresponding Hamilton–Jacobi–Bellman (HJB) equations are deduced for both the postdefault case and the predefault case. Using the variable change and variable separation techniques, we obtain the optimal reinsurance and investment strategies and the corresponding value functions. Our goal is to maximize the expected utility of terminal wealth under the worst-case scenario according to the max-min expected utility. Finally, we exemplify our deductions by some special cases and numerical cases, which verify our theoretical results.Here, we arrange the remaining part of this paper as follows: Section2 formulates the robust investment-reinsurance optimization regarding the default risk under the jump-diffusion model. In Section 3, we derive the closed-form expressions for the optimal strategies and the corresponding value functions under the predefault case and postdefault case, respectively. Section 4 provides a proof of the verification theorem. Section 5 provides some special cases. Numerical examples of our results are demonstrated in Section 6. Finally, conclusions are given in Section 7.


## 2. Model Formulation

In this article, we consider a complete probability spaceΩ,ℱ,P. Let F:=ℱtt≥0 be the right-continuous, P-complete filtration generated by two standard Brownian motions W1t and W2t, a Poisson process Nt, and two families of random variables Yi,i≥1 and Zi,i≥1. We assume that W1t,W2t,Nt, Yi, and Zi are mutually independent. Let G:=Gtt≥0 be the enlarged filtration of F and ℍ, i.e., Gt:=ℱt∨ℱtH, where the filtration ℍ=ℱtHt≥0 is generated by a default process Ht. We assume that every F-martingale is also a G-martingale. The probability measure P is the real-world probability measure, and Q is the risk-neutral measure.


### 2.1. The Financial Market

In this section, we consider a financial market consisting of three types of securities: a risk-free asset, a stock, and a defaultable corporate bond. The price process of the risk-free asset under the measureP is described by(1)dRt=rRtdt,where r>0 is the risk-free interest rate. The price process Stt≥0 of a stock is described by a jump-diffusion process:(2)dSt=St−μdt+σdW1t+d∑i=1N1tYi,where μ>0 is the expected instantaneous rate of return of the stock; σ is a positive constant; N1tt∈0,T is a homogeneous Poisson process with intensity λ1, representing the number of a stock price’s jumps during the time interval 0,t; and Yi is the ith jump amplitude of the stock’s price, and Y1,Y2,…, are i.i.d. random variables with the common distribution function F1y, the first moment EYi=μY, and the second moment EYi2=σY2. We assume that PYi>−1foralli≥1 to ensure that the stock’s price remains positive. Generally, the expect return of the stock is larger than the risk-free interest rate, so we assume that μ+λ1μY>r.Next, we consider thatNdt,dy is a Poisson random measure on Ω×0,T×−1,∞ and υdt,dy is the compensator measure of Ndt,dy. That is, N˜dt,dy=Ndt,dy−υdt,dy, and then(3)dSt=St−μdt+σdW1t+∫−1∞yNdt,dy,where υdt,dy=λ1dtdF1y. By the definition of F, for any A∈ℬ−1,∞, N˜t,At∈0,T is a P,F-martingale.Next, we consider the price process of the defaultable corporate bond by the intensity-based approach. Letτ be the time of default and τ represent the first jump time of a Poisson process with constant intensity hP>0 under P. A default indicator process is defined as Ht:=Iτ≤t for each t≥0, and the value of the corporate bond is assumed to be zero after default. Let ℍ be the filtration generated by the default process Ht and augmented in the usual way. By definition, τ is naturally an ℍ-stopping time and a G-stopping time. Furthermore, the martingale default process is thus given by MPt=Ht−∫0t1−HuhPdu, which is a P,G-martingale. By Girsanov’s theorem in Bielecki and Jang [15], under the chosen risk-neutral measure Q, the arrival intensity of default is given by hQ=hP/Δ. We denote that 1/Δ≥1 is the default risk premium. We assume that there exists a defaultable zero coupon bond with a maturity date T1, and the insurer can recover a fraction of the market value of the defaultable bond just prior to default. Now, for the positive interest rate r, the price dynamics of the defaultable bond under P is (see Deng et al. [21])(4)dpt,T1=pt−,T1rdt+1−Htδ1−Δdt−1−Ht−ζdMPt,where δ:=ζhQ represents the risk-neutral credit spread and 0<ζ<1 denotes the loss rate of the bond when a default occurs.


### 2.2. Dynamics of Surplus Process

The insurer’s surplus processU0tt≥0 is described by a jump-diffusion risk model:(5)dU0t=cdt+σ0dW2t−d∑i=1N2tZi,where c is the premium rate and σ0≥0 is a constant. ∑i=1N2tZi represents the aggregate claim amount up to time t, where N2t is a homogeneous Poisson process with intensity λ2>0, and the individual claim sizes Z1,Z2,…, independent of N2t, are i.i.d. positive random variables with the common distribution function F2z, the first moment EZi=μZ, and the second moment EZi2=σZ2.In addition, the insurance premium ratec under the expected value principle is given by c=λ2μZ1+θ0, where θ0>0 is the relative safety loading of the insurer. Suppose the insurer wants to reduce his/her risk by purchasing reinsurance. If there is a claim Zi at time t, a proportion ℋZi (self-reinsurance function, see Schmidli [39]) is paid by the insurer, and the rest Zi−ℋZi is paid by the reinsurer. cℋ is the premium rate of the reinsurer and is calculated according to the generalized mean-variance principle [5]. So, the premium rate of the insurer is(6)c−cℋ=λ2θ0−ημZ+1+ηEℋZi−θ1EZi−ℋZi2,where η≥0 and θ1≥0 is the relative safety loading of the reinsurer.According to Grandell [40], the surplus process can be approximated by the following diffusion process:(7)dUt=λ2θ0−ημZ+ηEℋZi−1+ηθ1EZi−ℋZi2dt+σ0dW2t+λ2EℋZi2dW3t.While the self-reinsurance functionℋ⋅ may take various forms, Zhang et al. [5] supposed that(8)ℋZi=aη+21+ηθ1Zi∧Zi,t∈0,T,where a∈0,1/21+ηθ1. Next, we will only consider reinsurance strategies given by (8). In this case, since ℋZi is uniquely characterized by the parameter a, we also rewrite ℋZi as ℋa,Zi to emphasize the dependence on a and call a as the insurer’s reinsurance strategy. That is, ℋa,Zi=aη+21+ηθ1Zi∧Zi. At any time t, with a larger a, the insurer reduces expenses on reinsurance and pays a larger proportion of each claim by himself/herself. Specially, when a=1/21+ηθ1, ℋa,Zi=Zi, that is, the insurer pays all of the claims by himself/herself; when a=0, he/she transfers all of the claims to the reinsurer according to Chen et al. [41]. Then, the surplus process of the insurer under the retention ℋ at time t is given by(9)dUℋt=λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2dt+σ0dW2t+λ2Eℋa,Zi2dW3t.Remark 1.
Ifη=0, then ℋZi=2aθ1Zi becomes a proportional reinsurance type (see also Zhou et al. [24] and Zheng et al. [26]); if θ1=0, then ℋZi=aη∧Zi becomes an excess-of-loss reinsurance type, the reinsurance premium under the mean principle (see also Zhao et al. [14] and Li et al. [10]). Therefore, proportional reinsurance and excess-of-loss reinsurance are special cases of (8).


### 2.3. The Wealth Process

In this section, we assume that the insurer is allowed to invest all his/her surplus in the financial market defined above. The insurer’s trading strategy isπt=π1t,π2t,a, where π1t is the total amount of wealth invested in the risky asset (a stock) at time t, π2t is the amount of wealth invested in the defaultable corporate bond, and a is the insurer’s reinsurance strategy. The remainder amount is invested in the risk-free asset. We assume that the corporate bond is not traded after default, and the investment horizon is 0,T, where T<T1. The reserve process subjected to this choice is denoted by XπT. Thus, the wealth process can be presented as follows:(10)dXπt=Xt−π1t−π2tRtdRt+π1tSt−dSt+π2tpt−,T1dpt,T1+dUℋt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2dt+π1tσdW1t+π1t∫−1∞yNdt,dy+σ0dW2t+λ2Eℋa,Zi2dW3t−π2t1−Ht−ζdHt.Suppose that the insurer has an exponential utility function defined by(11)Ux=−1αe−αx,α>0.For an admissible controlπt and an initial value x,h, we define the objective function as(12)supπ∈ΠEUXπTXt,Ht=x,h=supπ∈ΠE−1αe−αXπTXt,Ht=x,h.We denote the set of all admissible strategies byΠ. Then, we have the following definition for the set of admissible strategies.Definition 1.
A trading strategyπt=π1t,π2t,at is said to be admissible if(i)
πt is G-progressively measurable(ii)
EPϕ∫0Tπ12t+π22t+a2tdt<∞(iii)
∀πt,Xπt, the stochastic differential equation (10) has a pathwise unique solution Xπt with EPϕexp−αXπt<∞, where Pϕ is the chosen measure to describe the worst case and will be shown later
However, the AAI wants to guard himself/herself against worst-case scenarios. We assume that the knowledge of the AAI about ambiguity is described by probabilityP, namely, the reference probability (or model). But, he/she is skeptical about this reference model and hopes to consider alternative models, which are defined as a class of probability measures P:=PϕPϕ∼P equivalent to P (Anderson et al. [27] and Zeng et al. [6]). At first, we define a process ϕt,y=ϕ1t,ϕ2t,ϕ3t,ϕ4t,ϕ5t,φt,y such that(1)
ϕt,y is Gt-measurable, for each t∈0,T(2)
ϕ4t=ϕ4t,ω>0 and ϕ5tφt,y=ϕ5t,ωφt,y,ω>0 a.s. t,y,ω∈0,T×−1,∞×Ω(3)
∫0Tϕ12t+ϕ22t+ϕ32t+ϕ42tdt<∞ and ∫0T∫−1∞ϕ52tφ2t,ydF1ydt<∞, P-a.s.
The alternative measuresϕ1t,ϕ2t,ϕ3t,ϕ4t,ϕ5t, and φt,y are positive stochastic processes. We write Σ for the space of all such processes ϕ. Note that P is the probability measure associated with the reference model. For every ϕ∈Σ, each probability measure Pϕ∈P has a Radon–Nikodym derivative:(13)dPϕdPGT:=ΛϕT,with respect to P, where the process Λϕt is modelled by the stochastic differential equation (see Jin et al. [38])(14)Λϕt:=exp∫0tϕ1udW1u−12∫0tϕ12udu−∫0tϕ2udW2u−12∫0tϕ22udu−∫0tϕ3udW3u−12∫0tϕ32udu+∫0tlnϕ4udHu+hP∫0t1−ϕ4u1−Hudu+∫0t∫−1∞lnϕ5uφu,yNdu,dy+∫0t∫−1∞1−ϕ5uφu,yυdu,dy,with Λϕ0=1, P-a.s. By the Itô differentiation rule, we get(15)dΛϕt=Λϕt−−ϕ1tdW1t−ϕ2tdW2t−ϕ3tdW3t−1−ϕ4tdMPt−∫−1∞1−ϕ5tφt,yN˜dt,dy.
Note thatϕ5t and φt,y are positive stochastic processes, and φt,y satisfies the following relationship:(16)∫−1∞φt,ydF1y=1.
The distribution of the stock’s jump amplitude and the jump intensity is ambiguous, so the density function is not equal underP and Pϕ. Under the probability measure Pϕ, the jump intensity λ1 and the density function dF1y of the stock’s price process are changed into ϕ5tλ1 and φt,ydF1y in the alternative model. That is, λ1Pϕt=ϕ5tλ1, and N˜Pϕdt,dy:=Ndt,dy−ϕ5tφt,yλ1dtdF1y is a compensated Poisson random measure. For the default indicator process Ht:=Hτ≤t,t≥0, the intensity hP of the jumps becomes ϕ4thP, and the jump size is always equal to 1, so the jump size distribution is identical under P and Pϕ. For three standard Brownian motions, according to Girsanov’s theorem, dWiPϕt=dWit+ϕitdt,i=1,2,3. Thus, the wealth process under Pϕ becomes that(17)dXπt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2−ϕ1tπ1tσ−ϕ3tλ2Eℋa,Zi2dt−σ0ϕ2t+π1tσdW1Pϕt+σ0dW2Pϕt+λ2Eℋa,Zi2dW3Pϕt+π1t∫−1∞yNdt,dy−π2t1−Ht−ζdHt.
To simplify further analysis, we define the following functions:(18)g0x=ηx1−21+ηθ1x,gz=η+21+ηθ1z,h1a:=ηh3a−1+ηθ1h4a,h2a:=Eℋa,Z2=2∫0ax∫g0x∞g2zdF2zdx,h3a:=Eℋa,Z=∫0a∫g0x∞g2zdF2zdx,h4a:=EZ−ℋa,Z2=2∫a1/21+ηθ1∫g0x∞z−xgzgzdF2zdx.
Then, the dynamics of the wealth process underPϕ is(19)dXπt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+h1a−ϕ1tπ1tσ−σ0ϕ2t−ϕ3tλ2h2adt+π1tσdW1Pϕt+σ0dW2Pϕt+λ2h2adW3Pϕt+π1t∫−1∞yNdt,dy−π2t1−Ht−ζdHt.
Next, we assume that the insurer determines a robust portfolio strategy which is the best choice in some worst-case models as Anderson et al. [42]. The insurer penalizes any deviation from this reference model and the penalty increases with this deviation. Then, we use relative entropy to measure the deviation between the reference measure P and an alternative measure Pϕ. The increase in relative entropy from t to t+dt is shown by(20)12ϕ12tdt+12ϕ22tdt+12ϕ32tdt+ϕ4tlnϕ4t−ϕ4t+1hP1−hdt+∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1ydt,with h∈0,1. The increase is caused by three diffusion components and two jump components.
On the basis of Branger and Larsen [37], which allows the insurer’s ambiguity aversion with respect to the diffusion risk and jump risk to differ from each other, we can modify problem (12) and define the value function as(21)Vt,x,h:=supπ∈Πinfϕ∈∑Ex,hPϕ−1αe−αXπT+∫tTGu,Xπu,ϕudu,where Ex,hPϕ is calculated under the alternative measure Pϕ, the initial values of the processes are given by Xπt=x,Ht=h, and(22)Gt,Xπt,ϕt=ϕ12t2Ψ1t,Xπt,Ht+ϕ22t2Ψ2t,Xπt,Ht+ϕ32t2Ψ3t,Xπt,Ht+ϕ4tlnϕ4t−ϕ4t+1hP1−HtΨ4t,Xπt,Ht+∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1yΨ5t,Xπt,Ht.
The five terms in (22) are scaled by Ψ1≥0, Ψ2≥0, Ψ3≥0, Ψ4≥0, and Ψ5≥0, which are state-dependent. We follow Maenhout [29] and set(23)Ψ1t,x,h=−β1αVt,x,h,Ψ2t,x,h=−β2αVt,x,h,Ψ3t,x,h=−β3αVt,x,h,Ψ4t,x,h=−β4αVt,x,h,Ψ5t,x,h=−β5αVt,x,h,where βi≥0,i=1,2,3,4,5, is the ambiguity aversion coefficient with respect to three diffusion risks and two jump risks. The larger the values of Ψ1,Ψ2,Ψ3,Ψ4,andΨ5 are, the less a given deviation from the reference model is penalized, the less the faith of the insurer in the reference model, and the more the worst-case model will deviate from the reference model.


## 2.1. The Financial Market

In this section, we consider a financial market consisting of three types of securities: a risk-free asset, a stock, and a defaultable corporate bond. The price process of the risk-free asset under the measureP is described by(1)dRt=rRtdt,where r>0 is the risk-free interest rate. The price process Stt≥0 of a stock is described by a jump-diffusion process:(2)dSt=St−μdt+σdW1t+d∑i=1N1tYi,where μ>0 is the expected instantaneous rate of return of the stock; σ is a positive constant; N1tt∈0,T is a homogeneous Poisson process with intensity λ1, representing the number of a stock price’s jumps during the time interval 0,t; and Yi is the ith jump amplitude of the stock’s price, and Y1,Y2,…, are i.i.d. random variables with the common distribution function F1y, the first moment EYi=μY, and the second moment EYi2=σY2. We assume that PYi>−1foralli≥1 to ensure that the stock’s price remains positive. Generally, the expect return of the stock is larger than the risk-free interest rate, so we assume that μ+λ1μY>r.Next, we consider thatNdt,dy is a Poisson random measure on Ω×0,T×−1,∞ and υdt,dy is the compensator measure of Ndt,dy. That is, N˜dt,dy=Ndt,dy−υdt,dy, and then(3)dSt=St−μdt+σdW1t+∫−1∞yNdt,dy,where υdt,dy=λ1dtdF1y. By the definition of F, for any A∈ℬ−1,∞, N˜t,At∈0,T is a P,F-martingale.Next, we consider the price process of the defaultable corporate bond by the intensity-based approach. Letτ be the time of default and τ represent the first jump time of a Poisson process with constant intensity hP>0 under P. A default indicator process is defined as Ht:=Iτ≤t for each t≥0, and the value of the corporate bond is assumed to be zero after default. Let ℍ be the filtration generated by the default process Ht and augmented in the usual way. By definition, τ is naturally an ℍ-stopping time and a G-stopping time. Furthermore, the martingale default process is thus given by MPt=Ht−∫0t1−HuhPdu, which is a P,G-martingale. By Girsanov’s theorem in Bielecki and Jang [15], under the chosen risk-neutral measure Q, the arrival intensity of default is given by hQ=hP/Δ. We denote that 1/Δ≥1 is the default risk premium. We assume that there exists a defaultable zero coupon bond with a maturity date T1, and the insurer can recover a fraction of the market value of the defaultable bond just prior to default. Now, for the positive interest rate r, the price dynamics of the defaultable bond under P is (see Deng et al. [21])(4)dpt,T1=pt−,T1rdt+1−Htδ1−Δdt−1−Ht−ζdMPt,where δ:=ζhQ represents the risk-neutral credit spread and 0<ζ<1 denotes the loss rate of the bond when a default occurs.


## 2.2. Dynamics of Surplus Process

The insurer’s surplus processU0tt≥0 is described by a jump-diffusion risk model:(5)dU0t=cdt+σ0dW2t−d∑i=1N2tZi,where c is the premium rate and σ0≥0 is a constant. ∑i=1N2tZi represents the aggregate claim amount up to time t, where N2t is a homogeneous Poisson process with intensity λ2>0, and the individual claim sizes Z1,Z2,…, independent of N2t, are i.i.d. positive random variables with the common distribution function F2z, the first moment EZi=μZ, and the second moment EZi2=σZ2.In addition, the insurance premium ratec under the expected value principle is given by c=λ2μZ1+θ0, where θ0>0 is the relative safety loading of the insurer. Suppose the insurer wants to reduce his/her risk by purchasing reinsurance. If there is a claim Zi at time t, a proportion ℋZi (self-reinsurance function, see Schmidli [39]) is paid by the insurer, and the rest Zi−ℋZi is paid by the reinsurer. cℋ is the premium rate of the reinsurer and is calculated according to the generalized mean-variance principle [5]. So, the premium rate of the insurer is(6)c−cℋ=λ2θ0−ημZ+1+ηEℋZi−θ1EZi−ℋZi2,where η≥0 and θ1≥0 is the relative safety loading of the reinsurer.According to Grandell [40], the surplus process can be approximated by the following diffusion process:(7)dUt=λ2θ0−ημZ+ηEℋZi−1+ηθ1EZi−ℋZi2dt+σ0dW2t+λ2EℋZi2dW3t.While the self-reinsurance functionℋ⋅ may take various forms, Zhang et al. [5] supposed that(8)ℋZi=aη+21+ηθ1Zi∧Zi,t∈0,T,where a∈0,1/21+ηθ1. Next, we will only consider reinsurance strategies given by (8). In this case, since ℋZi is uniquely characterized by the parameter a, we also rewrite ℋZi as ℋa,Zi to emphasize the dependence on a and call a as the insurer’s reinsurance strategy. That is, ℋa,Zi=aη+21+ηθ1Zi∧Zi. At any time t, with a larger a, the insurer reduces expenses on reinsurance and pays a larger proportion of each claim by himself/herself. Specially, when a=1/21+ηθ1, ℋa,Zi=Zi, that is, the insurer pays all of the claims by himself/herself; when a=0, he/she transfers all of the claims to the reinsurer according to Chen et al. [41]. Then, the surplus process of the insurer under the retention ℋ at time t is given by(9)dUℋt=λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2dt+σ0dW2t+λ2Eℋa,Zi2dW3t.Remark 1.
Ifη=0, then ℋZi=2aθ1Zi becomes a proportional reinsurance type (see also Zhou et al. [24] and Zheng et al. [26]); if θ1=0, then ℋZi=aη∧Zi becomes an excess-of-loss reinsurance type, the reinsurance premium under the mean principle (see also Zhao et al. [14] and Li et al. [10]). Therefore, proportional reinsurance and excess-of-loss reinsurance are special cases of (8).


## 2.3. The Wealth Process

In this section, we assume that the insurer is allowed to invest all his/her surplus in the financial market defined above. The insurer’s trading strategy isπt=π1t,π2t,a, where π1t is the total amount of wealth invested in the risky asset (a stock) at time t, π2t is the amount of wealth invested in the defaultable corporate bond, and a is the insurer’s reinsurance strategy. The remainder amount is invested in the risk-free asset. We assume that the corporate bond is not traded after default, and the investment horizon is 0,T, where T<T1. The reserve process subjected to this choice is denoted by XπT. Thus, the wealth process can be presented as follows:(10)dXπt=Xt−π1t−π2tRtdRt+π1tSt−dSt+π2tpt−,T1dpt,T1+dUℋt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2dt+π1tσdW1t+π1t∫−1∞yNdt,dy+σ0dW2t+λ2Eℋa,Zi2dW3t−π2t1−Ht−ζdHt.Suppose that the insurer has an exponential utility function defined by(11)Ux=−1αe−αx,α>0.For an admissible controlπt and an initial value x,h, we define the objective function as(12)supπ∈ΠEUXπTXt,Ht=x,h=supπ∈ΠE−1αe−αXπTXt,Ht=x,h.We denote the set of all admissible strategies byΠ. Then, we have the following definition for the set of admissible strategies.Definition 1.
A trading strategyπt=π1t,π2t,at is said to be admissible if(i)
πt is G-progressively measurable(ii)
EPϕ∫0Tπ12t+π22t+a2tdt<∞(iii)
∀πt,Xπt, the stochastic differential equation (10) has a pathwise unique solution Xπt with EPϕexp−αXπt<∞, where Pϕ is the chosen measure to describe the worst case and will be shown later
However, the AAI wants to guard himself/herself against worst-case scenarios. We assume that the knowledge of the AAI about ambiguity is described by probabilityP, namely, the reference probability (or model). But, he/she is skeptical about this reference model and hopes to consider alternative models, which are defined as a class of probability measures P:=PϕPϕ∼P equivalent to P (Anderson et al. [27] and Zeng et al. [6]). At first, we define a process ϕt,y=ϕ1t,ϕ2t,ϕ3t,ϕ4t,ϕ5t,φt,y such that(1)
ϕt,y is Gt-measurable, for each t∈0,T(2)
ϕ4t=ϕ4t,ω>0 and ϕ5tφt,y=ϕ5t,ωφt,y,ω>0 a.s. t,y,ω∈0,T×−1,∞×Ω(3)
∫0Tϕ12t+ϕ22t+ϕ32t+ϕ42tdt<∞ and ∫0T∫−1∞ϕ52tφ2t,ydF1ydt<∞, P-a.s.
The alternative measuresϕ1t,ϕ2t,ϕ3t,ϕ4t,ϕ5t, and φt,y are positive stochastic processes. We write Σ for the space of all such processes ϕ. Note that P is the probability measure associated with the reference model. For every ϕ∈Σ, each probability measure Pϕ∈P has a Radon–Nikodym derivative:(13)dPϕdPGT:=ΛϕT,with respect to P, where the process Λϕt is modelled by the stochastic differential equation (see Jin et al. [38])(14)Λϕt:=exp∫0tϕ1udW1u−12∫0tϕ12udu−∫0tϕ2udW2u−12∫0tϕ22udu−∫0tϕ3udW3u−12∫0tϕ32udu+∫0tlnϕ4udHu+hP∫0t1−ϕ4u1−Hudu+∫0t∫−1∞lnϕ5uφu,yNdu,dy+∫0t∫−1∞1−ϕ5uφu,yυdu,dy,with Λϕ0=1, P-a.s. By the Itô differentiation rule, we get(15)dΛϕt=Λϕt−−ϕ1tdW1t−ϕ2tdW2t−ϕ3tdW3t−1−ϕ4tdMPt−∫−1∞1−ϕ5tφt,yN˜dt,dy.
Note thatϕ5t and φt,y are positive stochastic processes, and φt,y satisfies the following relationship:(16)∫−1∞φt,ydF1y=1.
The distribution of the stock’s jump amplitude and the jump intensity is ambiguous, so the density function is not equal underP and Pϕ. Under the probability measure Pϕ, the jump intensity λ1 and the density function dF1y of the stock’s price process are changed into ϕ5tλ1 and φt,ydF1y in the alternative model. That is, λ1Pϕt=ϕ5tλ1, and N˜Pϕdt,dy:=Ndt,dy−ϕ5tφt,yλ1dtdF1y is a compensated Poisson random measure. For the default indicator process Ht:=Hτ≤t,t≥0, the intensity hP of the jumps becomes ϕ4thP, and the jump size is always equal to 1, so the jump size distribution is identical under P and Pϕ. For three standard Brownian motions, according to Girsanov’s theorem, dWiPϕt=dWit+ϕitdt,i=1,2,3. Thus, the wealth process under Pϕ becomes that(17)dXπt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+ηEℋa,Zi−1+ηθ1EZi−ℋa,Zi2−ϕ1tπ1tσ−ϕ3tλ2Eℋa,Zi2dt−σ0ϕ2t+π1tσdW1Pϕt+σ0dW2Pϕt+λ2Eℋa,Zi2dW3Pϕt+π1t∫−1∞yNdt,dy−π2t1−Ht−ζdHt.
To simplify further analysis, we define the following functions:(18)g0x=ηx1−21+ηθ1x,gz=η+21+ηθ1z,h1a:=ηh3a−1+ηθ1h4a,h2a:=Eℋa,Z2=2∫0ax∫g0x∞g2zdF2zdx,h3a:=Eℋa,Z=∫0a∫g0x∞g2zdF2zdx,h4a:=EZ−ℋa,Z2=2∫a1/21+ηθ1∫g0x∞z−xgzgzdF2zdx.
Then, the dynamics of the wealth process underPϕ is(19)dXπt=rXt+μ−rπ1t+π2t1−Htδ+λ2θ0−ημZ+h1a−ϕ1tπ1tσ−σ0ϕ2t−ϕ3tλ2h2adt+π1tσdW1Pϕt+σ0dW2Pϕt+λ2h2adW3Pϕt+π1t∫−1∞yNdt,dy−π2t1−Ht−ζdHt.
Next, we assume that the insurer determines a robust portfolio strategy which is the best choice in some worst-case models as Anderson et al. [42]. The insurer penalizes any deviation from this reference model and the penalty increases with this deviation. Then, we use relative entropy to measure the deviation between the reference measure P and an alternative measure Pϕ. The increase in relative entropy from t to t+dt is shown by(20)12ϕ12tdt+12ϕ22tdt+12ϕ32tdt+ϕ4tlnϕ4t−ϕ4t+1hP1−hdt+∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1ydt,with h∈0,1. The increase is caused by three diffusion components and two jump components.
On the basis of Branger and Larsen [37], which allows the insurer’s ambiguity aversion with respect to the diffusion risk and jump risk to differ from each other, we can modify problem (12) and define the value function as(21)Vt,x,h:=supπ∈Πinfϕ∈∑Ex,hPϕ−1αe−αXπT+∫tTGu,Xπu,ϕudu,where Ex,hPϕ is calculated under the alternative measure Pϕ, the initial values of the processes are given by Xπt=x,Ht=h, and(22)Gt,Xπt,ϕt=ϕ12t2Ψ1t,Xπt,Ht+ϕ22t2Ψ2t,Xπt,Ht+ϕ32t2Ψ3t,Xπt,Ht+ϕ4tlnϕ4t−ϕ4t+1hP1−HtΨ4t,Xπt,Ht+∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1yΨ5t,Xπt,Ht.
The five terms in (22) are scaled by Ψ1≥0, Ψ2≥0, Ψ3≥0, Ψ4≥0, and Ψ5≥0, which are state-dependent. We follow Maenhout [29] and set(23)Ψ1t,x,h=−β1αVt,x,h,Ψ2t,x,h=−β2αVt,x,h,Ψ3t,x,h=−β3αVt,x,h,Ψ4t,x,h=−β4αVt,x,h,Ψ5t,x,h=−β5αVt,x,h,where βi≥0,i=1,2,3,4,5, is the ambiguity aversion coefficient with respect to three diffusion risks and two jump risks. The larger the values of Ψ1,Ψ2,Ψ3,Ψ4,andΨ5 are, the less a given deviation from the reference model is penalized, the less the faith of the insurer in the reference model, and the more the worst-case model will deviate from the reference model.


## 3. The Main Result

In this section, the goal is to find the optimal allocation pairπ1t,π2t,at under the worst-case scenario. According to the dynamic programming principle, the HJB equation can be derived as (Anderson et al. [42])(24)supπ∈Πinfϕ∈ΣAπ,ϕV+Gt,Xπt,ϕt=0,with the boundary condition VT,x,h=−1/αe−αx, where Aπ,ϕ is the infinitesimal generator of (21) under Pϕ and is defined by(25)Aπ,ϕVt,x,h=Vt+Vxrx+μ−rπ1t+1−hδπ2t+λ2θ0−ημZ+h1a−σϕ1tπ1t−σ0ϕ2t−ϕ3tλ2h2a+12Vxxσ2π12t+σ02+λ2h2a+λ1EPϕVt,x+π1tY,h−Vt,x,h+hP1−hVt,x−π2tζ,1−Vt,x,0ϕ4t,where Vt,Vx,andVxx represent the value function’s partial derivatives with respect to the corresponding variables. We split equation (24) into two cases: the postdefault case (h=1) and the predefault case (h=0), and denote V as follows:(26)Vt,x,h=Vt,x,1=Vt,x,0.According to (26) and (21), the HJB equation (24) transforms into the following two forms: (27)0=supπ∈Πinfϕ∈ΣVtt,x,1+Vxt,x,1rx+μ−rπ1t+λ2θ0−ημZ+h1a−σϕ1tπ1t−σ0ϕ2t−ϕ3tλ2h2a+12Vxxt,x,1σ2π12t+σ02+λ2h2a+EPϕVt,x+π1tY,1−Vt,x,1−ϕ12tα2β1Vt,x,1−ϕ22tα2β2Vt,x,1−ϕ32tα2β3Vt,x,1−∫−1∞ϕ5φt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1yβ5αVt,x,1,(28)0=supπ∈Πinfϕ∈ΣVtt,x,0+Vxt,x,0rx+μ−rπ1t+δπ2t+λ2θ0−ημZ+h1a−σϕ1tπ1t−σ0ϕ2t−ϕ3tλ2h2a+12Vxxt,x,0σ2π12t+σ02+λ2h2a+EPϕVt,x+π1tY,0−Vt,x,0+hPϕ5tVt,x−π2tζ,1−Vt,x,0−ϕ12t2β1αVt,x,0−ϕ22t2β2αVt,x,0−ϕ32t2β3αVt,x,0−∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1λ1dF1yβ5αVt,x,0−ϕ4tlnϕ4t−ϕ4t+1β4hPαVt,x,0.In the following two sections, we derive the optimal reinsurance and investment strategies and corresponding value functions in the postdefault case and predefault case, respectively.


### 3.1. The Postdefault Case

In this section, we will concentrate on the postdefault case, that is, HJB equation (27) for Vt,x,1, and we conjecture that the value function has the following form:(29)Vt,x,1=−1αexp−αxerT−tg1t,where g1t is a deterministic function, with g1T=1. We get(30)Vtt,x,1=g1′tg1t+rαxerT−tVt,x,1,Vxt,x,1=−αerT−tVt,x,1,Vxxt,x,1=α2e2rT−tVt,x,1,EVt,x+π1tY,1−Vt,x,1=λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1yVt,x,1.Substituting these partial derivatives andE⋅ into equation (27), we get(31)0=infϕ∈Σsupπ∈Πg1′tg1t−αerT−tμ−rπ1t−σϕ1tπ1t−σ0ϕ2t+λ2θ0−ημZ+h1a−ϕ3tλ2h2a+12α2e2rT−tσ2π12t+σ02+12α2e2rT−tλ2h2a+λ1∫−1∞e−απ1terT−ty−1ϕ4tφt,ydF1y−ϕ12t2β1α−ϕ22t2β2α−ϕ32t2β3α−λ1αβ5∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1dF1y.Fixingπ and a and maximizing over ϕ yield the following first-order condition for the minimum point ϕ∗=ϕ1∗,ϕ2∗,ϕ3∗,ϕ5∗,φ∗ (there is no ambiguity about the default jump risk after default):(32)ϕ1∗t=β1σπ1terT−t,ϕ2∗t=β2σ0erT−t,ϕ3∗t=β3λ2h2aerT−t,ϕ5∗tφ∗t,y=expβ5αe−απ1terT−ty−1.Noting thatEφt,Y=1 (formula (16)), we have (33)φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1.Lemma 1.
For anyt∈0,T, the equation (34)π1t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y,has a unique positive solution π1∗t.Proof.
SupposeWt,π1=μ−r+λ1∫−1∞ye−απ1terT−tyexpβ5/αe−απ1terT−ty−1dF1y−σ2α+β1π1terT−t, then we get(35)∂Wt,π1∂π1=−λ1αerT−t∫−1∞y2e−απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y−σ2α+β1erT−t−λ1β5erT−t∫−1∞y2e−2απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y<0,which implies that Wt,π1 is a decreasing function w.r.t. π1. Furthermore, we have Wt,0=μ−r+λ1∫−1∞ydF1y=μ−r+λ1μY>0. Also, we can find that if π1t>μ−r+λ1μY/σ2α+β1erT−t>0, we have Wt,π1<0. Therefore, equation (34) has a unique positive root π1∗t.Lemma 2.
For the functionshia, i=1,2,3,4, defined above, we have the following relationship:(36)h1′a=−h2′a221+ηθ1−1a,h1a=−1+ηθ1h40−h2a221+ηθ1−1a+∫0ah2x2x2dx,=−1+ηθ1h40−1+ηθ1h2a+∫0a∫g0x∞gzdF2zdx.Substitutingϕ∗ into (31), according to the first-order condition and Lemmas 1 and 2, we can obtain the maximum point π∗:=π1∗,π2∗,a∗ given by(37)π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=0,a∗t=121+ηθ1+α+β3erT−t.Substitutingπ∗ and ϕ∗ into (31), we get the equation(38)0=g1′tg1t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−λ1αβ5−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−αλ2erT−th1a∗t−α+β32h2a∗terT−t.Therefore, we can derive the following theorem.Theorem 1 (postdefault strategy).
The robust optimal reinsurance and investment strategies for the period after default are given as(39)ϕ1∗t=β1σπ1terT−t,ϕ2∗t=β2σ0erT−t,ϕ3∗t=β3λ2h2aerT−t,φ∗t,y=1ϕ5∗expβ5αe−απ1terT−ty−1,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=0,a∗t=121+ηθ1+α+β3erT−t.Furthermore, the postdefault value function is given by(40)Vt,x,1=−1αexp−αxerT−tg1t,where(41)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−udu.


### 3.2. The Predefault Case

In this section, we will concentrate on the predefault case, that is, HJB equation (28) for Vt,x,0, and we conjecture that the value function has the following form:(42)Vt,x,0=−1αexp−αxerT−tg2t,where g2t is a deterministic function, with g2T=1. We get(43)Vtt,x,0=g2′tg2t+rαxerT−tVt,x,0,Vxt,x,0=−αerT−tVt,x,0,Vxxt,x,0=α2e2rT−tVt,x,0,EVt,x+π1tY,0−Vt,x,0=λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1yVt,x,0,Vt,x−π2tζ,1−Vt,x,0=g1tg2texpαζπ2terT−t−1Vt,x,0.Substituting these partial derivatives,E⋅, and V⋅,⋅,1−V⋅,⋅,0 into equation (28), we get(44)0=infϕ∈Σsupπ∈Πg2′tg2t−αerT−tμ−rπ1t+π2tδ−σϕ1tπ1t−σ0ϕ2t+λ2θ0−ημZ+h1a−ϕ3tλ2h2a+12α2e2rT−tσ2π12t+σ02+λ2h2a−ϕ12t2β1α−ϕ22t2β2α−ϕ32t2β3α+λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1y+ϕ5thPg1tg2texpαζπ2terT−t−1−ϕ4tlnϕ4t−ϕ4t+1β4hPα−λ1αβ5∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1dF1y.Fixingπ and maximizing over ϕ yield the following first-order condition for the minimum point ϕ∗′=ϕ1∗′,ϕ2∗′,ϕ3∗′,ϕ4∗′,ϕ5∗′,φ∗′:(45)ϕ1∗′t=β1σπ1terT−t,ϕ2∗′t=β2σ0erT−t,ϕ3∗′t=β3λ2h2aerT−t,lnϕ4∗t=β4αg1tg2texpαζπ2terT−t−1,ϕ5∗′tφ∗′t,y=expβ5αe−απ1terT−ty−1.Noting thatEφt,Y=1, we have(46)φ∗′t,y=1ϕ5∗′texpβ5αe−απ1terT−ty−1,ϕ5∗′t=Eexpβ5αe−απ1terT−tY−1.Substitutingϕ∗′ into (44), according to the first-order condition, we can obtain the maximum point π∗′:=π1∗′,π2∗′,a∗′ given by (47)π1∗′t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗′terT−tyexpβ5αe−απ1∗′terT−ty−1dF1y,(48)π2∗′t=e−rT−tαζlng2tΔϕ4∗tg1t,(49)a∗′t=121+ηθ1+α+β3erT−t.Substitutingπ2∗′t into equation (45), we have αhP/β4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δ/ζ=0, and this equation has a unique positive root by the following lemma.Lemma 3.
LetW˜t,ϕ4=αhP/β4ϕ4tlnϕ4t+hPϕ4t−δ/ζ, then W˜t,ϕ4 has a unique positive root ϕ4∗t.Proof.
It is similar to the proof of Proposition 4.2 in the study of Sun et al. [23]. So we omit it.
According to (47) and (49), we obtain a∗′=a∗ and π1∗′=π1∗. Substituting π∗′ and ϕ∗′ into (44), we get the equation(50)0=g2′tg2t−δζlng2t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−λ1αβ5−αλ2erT−th1a∗t−α+β32h2a∗terT−t+δζlnΔϕ4∗t+lng1t+ϕ4∗t−1β4αΔ.
Note thatg2t>0; in order to get the expression for g2t with the boundary condition g2T=1, we try the following form of g2t=eg¯2t:(51)0=g¯2′t−δζg¯2t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−λ1αβ5−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−αλ2erT−th1a∗t−α+β32h2a∗terT−t+δζlnΔϕ4∗t+lng1t+ϕ4∗t−1β4αΔ.
Therefore, we can derive the following theorem.Theorem 2 (predefault strategy).
The robust optimal reinsurance and investment strategies for the period before default are given as(52)ϕ1∗′t=β1σπ1terT−t=ϕ1∗t,ϕ2∗′t=β2σ0erT−t=ϕ2∗t,ϕ3∗′t=β3λ2h2aerT−t=ϕ3∗t,αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ=0,φ∗′t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1=φ∗t,y,ϕ5∗′t=Eexpβ5αe−απ1terT−tY−1=ϕ5∗t,π1∗′t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗′terT−tyexpβ5αe−απ1∗′terT−ty−1dF1y=π1∗t,π2∗′t=e−rT−tαζlng2tΔϕ4∗tg1t,a∗′t=121+ηθ1+α+β3erT−t=a∗t.Furthermore, the predefault value function is given by(53)Vt,x,0=−1αexp−αxerT−tg2t,where g2t=eg¯2t, in which(54)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αerT−uλ2θ0−ημZ+μ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Next, putting the predefault and postdefault cases together, we have the following solution to the HJB equation (24) associated with the value function Vt,x,h:(55)Vt,x,h=1−hVt,x,0+hVt,x,1,whereh=0or1.Let us define the following processes which are candidate optimal strategies:(56)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=β3λ2h2aerT−t,t∈0,T,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,T,0,t∈τ∧T,T,a∗t=121+ηθ1+α+β3erT−t,t∈0,T.From the above expressions forϕ1∗,ϕ2∗,ϕ3∗,ϕ4∗,ϕ5∗,andφ∗, it is easy to verify that the expression Λϕt with ϕ1∗,ϕ2∗,ϕ3∗,ϕ4∗,ϕ5∗,andφ∗ instead of ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,andφ is indeed a P-martingale, which ensures a well-defined Pϕ∗.In the next section, we shall show that the above stochastic control policies are indeed the optimal strategies and that the value functionV^t,x,h is unique.


## 3.1. The Postdefault Case

In this section, we will concentrate on the postdefault case, that is, HJB equation (27) for Vt,x,1, and we conjecture that the value function has the following form:(29)Vt,x,1=−1αexp−αxerT−tg1t,where g1t is a deterministic function, with g1T=1. We get(30)Vtt,x,1=g1′tg1t+rαxerT−tVt,x,1,Vxt,x,1=−αerT−tVt,x,1,Vxxt,x,1=α2e2rT−tVt,x,1,EVt,x+π1tY,1−Vt,x,1=λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1yVt,x,1.Substituting these partial derivatives andE⋅ into equation (27), we get(31)0=infϕ∈Σsupπ∈Πg1′tg1t−αerT−tμ−rπ1t−σϕ1tπ1t−σ0ϕ2t+λ2θ0−ημZ+h1a−ϕ3tλ2h2a+12α2e2rT−tσ2π12t+σ02+12α2e2rT−tλ2h2a+λ1∫−1∞e−απ1terT−ty−1ϕ4tφt,ydF1y−ϕ12t2β1α−ϕ22t2β2α−ϕ32t2β3α−λ1αβ5∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1dF1y.Fixingπ and a and maximizing over ϕ yield the following first-order condition for the minimum point ϕ∗=ϕ1∗,ϕ2∗,ϕ3∗,ϕ5∗,φ∗ (there is no ambiguity about the default jump risk after default):(32)ϕ1∗t=β1σπ1terT−t,ϕ2∗t=β2σ0erT−t,ϕ3∗t=β3λ2h2aerT−t,ϕ5∗tφ∗t,y=expβ5αe−απ1terT−ty−1.Noting thatEφt,Y=1 (formula (16)), we have (33)φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1.Lemma 1.
For anyt∈0,T, the equation (34)π1t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y,has a unique positive solution π1∗t.Proof.
SupposeWt,π1=μ−r+λ1∫−1∞ye−απ1terT−tyexpβ5/αe−απ1terT−ty−1dF1y−σ2α+β1π1terT−t, then we get(35)∂Wt,π1∂π1=−λ1αerT−t∫−1∞y2e−απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y−σ2α+β1erT−t−λ1β5erT−t∫−1∞y2e−2απ1terT−tyexpβ5αe−απ1terT−ty−1dF1y<0,which implies that Wt,π1 is a decreasing function w.r.t. π1. Furthermore, we have Wt,0=μ−r+λ1∫−1∞ydF1y=μ−r+λ1μY>0. Also, we can find that if π1t>μ−r+λ1μY/σ2α+β1erT−t>0, we have Wt,π1<0. Therefore, equation (34) has a unique positive root π1∗t.Lemma 2.
For the functionshia, i=1,2,3,4, defined above, we have the following relationship:(36)h1′a=−h2′a221+ηθ1−1a,h1a=−1+ηθ1h40−h2a221+ηθ1−1a+∫0ah2x2x2dx,=−1+ηθ1h40−1+ηθ1h2a+∫0a∫g0x∞gzdF2zdx.Substitutingϕ∗ into (31), according to the first-order condition and Lemmas 1 and 2, we can obtain the maximum point π∗:=π1∗,π2∗,a∗ given by(37)π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=0,a∗t=121+ηθ1+α+β3erT−t.Substitutingπ∗ and ϕ∗ into (31), we get the equation(38)0=g1′tg1t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−λ1αβ5−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−αλ2erT−th1a∗t−α+β32h2a∗terT−t.Therefore, we can derive the following theorem.Theorem 1 (postdefault strategy).
The robust optimal reinsurance and investment strategies for the period after default are given as(39)ϕ1∗t=β1σπ1terT−t,ϕ2∗t=β2σ0erT−t,ϕ3∗t=β3λ2h2aerT−t,φ∗t,y=1ϕ5∗expβ5αe−απ1terT−ty−1,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=0,a∗t=121+ηθ1+α+β3erT−t.Furthermore, the postdefault value function is given by(40)Vt,x,1=−1αexp−αxerT−tg1t,where(41)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−udu.


## 3.2. The Predefault Case

In this section, we will concentrate on the predefault case, that is, HJB equation (28) for Vt,x,0, and we conjecture that the value function has the following form:(42)Vt,x,0=−1αexp−αxerT−tg2t,where g2t is a deterministic function, with g2T=1. We get(43)Vtt,x,0=g2′tg2t+rαxerT−tVt,x,0,Vxt,x,0=−αerT−tVt,x,0,Vxxt,x,0=α2e2rT−tVt,x,0,EVt,x+π1tY,0−Vt,x,0=λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1yVt,x,0,Vt,x−π2tζ,1−Vt,x,0=g1tg2texpαζπ2terT−t−1Vt,x,0.Substituting these partial derivatives,E⋅, and V⋅,⋅,1−V⋅,⋅,0 into equation (28), we get(44)0=infϕ∈Σsupπ∈Πg2′tg2t−αerT−tμ−rπ1t+π2tδ−σϕ1tπ1t−σ0ϕ2t+λ2θ0−ημZ+h1a−ϕ3tλ2h2a+12α2e2rT−tσ2π12t+σ02+λ2h2a−ϕ12t2β1α−ϕ22t2β2α−ϕ32t2β3α+λ1∫−1∞e−απ1terT−ty−1ϕ5tφt,ydF1y+ϕ5thPg1tg2texpαζπ2terT−t−1−ϕ4tlnϕ4t−ϕ4t+1β4hPα−λ1αβ5∫−1∞ϕ5tφt,ylnϕ5tφt,y−ϕ5tφt,y+1dF1y.Fixingπ and maximizing over ϕ yield the following first-order condition for the minimum point ϕ∗′=ϕ1∗′,ϕ2∗′,ϕ3∗′,ϕ4∗′,ϕ5∗′,φ∗′:(45)ϕ1∗′t=β1σπ1terT−t,ϕ2∗′t=β2σ0erT−t,ϕ3∗′t=β3λ2h2aerT−t,lnϕ4∗t=β4αg1tg2texpαζπ2terT−t−1,ϕ5∗′tφ∗′t,y=expβ5αe−απ1terT−ty−1.Noting thatEφt,Y=1, we have(46)φ∗′t,y=1ϕ5∗′texpβ5αe−απ1terT−ty−1,ϕ5∗′t=Eexpβ5αe−απ1terT−tY−1.Substitutingϕ∗′ into (44), according to the first-order condition, we can obtain the maximum point π∗′:=π1∗′,π2∗′,a∗′ given by (47)π1∗′t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗′terT−tyexpβ5αe−απ1∗′terT−ty−1dF1y,(48)π2∗′t=e−rT−tαζlng2tΔϕ4∗tg1t,(49)a∗′t=121+ηθ1+α+β3erT−t.Substitutingπ2∗′t into equation (45), we have αhP/β4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δ/ζ=0, and this equation has a unique positive root by the following lemma.Lemma 3.
LetW˜t,ϕ4=αhP/β4ϕ4tlnϕ4t+hPϕ4t−δ/ζ, then W˜t,ϕ4 has a unique positive root ϕ4∗t.Proof.
It is similar to the proof of Proposition 4.2 in the study of Sun et al. [23]. So we omit it.
According to (47) and (49), we obtain a∗′=a∗ and π1∗′=π1∗. Substituting π∗′ and ϕ∗′ into (44), we get the equation(50)0=g2′tg2t−δζlng2t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−λ1αβ5−αλ2erT−th1a∗t−α+β32h2a∗terT−t+δζlnΔϕ4∗t+lng1t+ϕ4∗t−1β4αΔ.
Note thatg2t>0; in order to get the expression for g2t with the boundary condition g2T=1, we try the following form of g2t=eg¯2t:(51)0=g¯2′t−δζg¯2t+α+β22ασ02e2rT−t−αλ2erT−tθ0−ημZ−λ1αβ5−αerT−tμ−rπ1∗t+α+β12ασ2e2rT−tπ1∗t2+λ1αβ5∫−1∞expβ5αe−απ1∗terT−ty−1dF1y−αλ2erT−th1a∗t−α+β32h2a∗terT−t+δζlnΔϕ4∗t+lng1t+ϕ4∗t−1β4αΔ.
Therefore, we can derive the following theorem.Theorem 2 (predefault strategy).
The robust optimal reinsurance and investment strategies for the period before default are given as(52)ϕ1∗′t=β1σπ1terT−t=ϕ1∗t,ϕ2∗′t=β2σ0erT−t=ϕ2∗t,ϕ3∗′t=β3λ2h2aerT−t=ϕ3∗t,αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ=0,φ∗′t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1=φ∗t,y,ϕ5∗′t=Eexpβ5αe−απ1terT−tY−1=ϕ5∗t,π1∗′t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗′terT−tyexpβ5αe−απ1∗′terT−ty−1dF1y=π1∗t,π2∗′t=e−rT−tαζlng2tΔϕ4∗tg1t,a∗′t=121+ηθ1+α+β3erT−t=a∗t.Furthermore, the predefault value function is given by(53)Vt,x,0=−1αexp−αxerT−tg2t,where g2t=eg¯2t, in which(54)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αerT−uλ2θ0−ημZ+μ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Next, putting the predefault and postdefault cases together, we have the following solution to the HJB equation (24) associated with the value function Vt,x,h:(55)Vt,x,h=1−hVt,x,0+hVt,x,1,whereh=0or1.Let us define the following processes which are candidate optimal strategies:(56)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=β3λ2h2aerT−t,t∈0,T,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,T,0,t∈τ∧T,T,a∗t=121+ηθ1+α+β3erT−t,t∈0,T.From the above expressions forϕ1∗,ϕ2∗,ϕ3∗,ϕ4∗,ϕ5∗,andφ∗, it is easy to verify that the expression Λϕt with ϕ1∗,ϕ2∗,ϕ3∗,ϕ4∗,ϕ5∗,andφ∗ instead of ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,andφ is indeed a P-martingale, which ensures a well-defined Pϕ∗.In the next section, we shall show that the above stochastic control policies are indeed the optimal strategies and that the value functionV^t,x,h is unique.


## 4. Verification Theorem

In order to verify the candidate optimal strategiesπ∗ and ϕ∗ are indeed optimal, and the candidate value function is (55), we give the verification theorem as follows.Proposition 1.
LetO:=0,T×R×0,1 and O¯ denote the closure of O. Suppose that there exist a function V^∈C1,2,1O∩CO¯ and a control ϕ∗,π∗∈Σ×Π such that(1)
Aϕ,π∗V^t,Xπ∗t,Ht+Gt,Xπ∗t,ϕt≥0, for all ϕ∈Σ(2)
Aϕ∗,πV^t,Xπt,Ht+Gt,Xπt,ϕ∗t≤0, for all π∈Π(3)
Aϕ∗,π∗V^t,Xπ∗t,Ht+Gt,Xπ∗t,ϕ∗t=0(4)
For allϕ,π∈Σ×Π: limt⟶T−V^t,Xπt,Ht=UXπT(5)
V^τ,Xπτ,Hττ∈T and Gτ,Xπτ,ϕττ∈T are uniformly integrable, where T denotes the set of stopping times τ≤T
Then,V^t,x,h=Vt,x,h and ϕ∗,π∗ are an optimal control.Proof.
According toV^∈C1,2,1O∩CO¯, choose ϕ,π∈Σ×Π, by the definition of Ht, V^τn,Xπτn,Hτn=V^τn,Xπτn,1, and V^τn,Xπτn,Hτn=V^τn,Xπτn,0; therefore, the Itô formula can be applied to V^τn,Xπτn,0:(57)V^τn,Xπτn,0=V^t,x,0+∫tτnAϕ,πV^u,Xπu,0du+∫tτnV^xπ1uσdW1Pϕu+∫tτnV^xσ0dW2Pϕu+∫tτnV^xλ2h2adW3Pϕu+∫tτnV^∫−1∞π1uyNdu,dy,where τn=T∧n∧infu>t;Xπu≥n, for n=1,2,…, and V^x=V^xt,x,0 and a=at for short. Because the continuous function V^x is bounded on the set t,τn×R, we obtain the following estimate with an appropriate constant M>0:(58)∫tτn12V^x2π12uσ2du≤M∫tτnπ12uσ2du.
Due to (ii) of Definition1, it follows that(59)Et,x,0Pϕ∫tτn12V^x2π12uσ2du<∞.
By the similar way as that in (59), we have(60)Et,x,0Pϕ∫tτn12V^x2σ02du<∞,Et,x,0Pϕ∫tτn12V^x2λ2h2adu<∞,Et,x,0Pϕ∫tτn12V^x2∫−1∞π1uyNdu,dy<∞.
Therefore, taking expectations in (57) leads to(61)V^t,x,0=Et,x,0PϕV^τn,Xπτn,0−∫tτnAϕ,πV^u,Xπu,0du.
If we apply (61) to ϕ,π∗ with ϕ∈Σ and use property 1, we get(62)V^t,x,0≤Et,x,0PϕV^τn,Xπ∗τn,0+∫tτnGu,Xπ∗u,ϕudu.
Lettingn⟶∞ and using properties 4 and 5, we have(63)V^t,x,0≤Et,x,0PϕUXπ∗T+∫tTGu,Xπ∗u,ϕudu.
Since this holds for allϕ∈Σ, we deduce that(64)V^t,x,0≤infϕ∈ΣEt,x,0PϕUXπ∗T+∫tTGu,Xπ∗u,ϕu.
Hence,(65)V^t,x,0≤supπ∈Πinfϕ∈ΣEt,x,0PϕUXπT+∫tτnGu,Xπu,ϕudu=Vt,x,0.
Next, if we apply (61) to ϕ∗,π with π∈Π and use property 2, we get(66)V^t,x,0≥Et,x,0PϕV^τn,Xπτn,0+∫tτnGu,Xπu,ϕ∗udu.
Lettingn⟶∞ and using properties 4 and 5, we have(67)V^t,x,0≥Et,x,0PϕUXπT+∫tTGu,Xπu,ϕ∗udu≥infϕ∈ΣEt,x,0PϕUXπT+∫tTGu,Xπu,ϕudu.
Since this holds for allπ∈Π, we deduce that(68)V^t,x,0≥supπ∈Πinfϕ∈ΣEt,x,0PϕUXπT+∫tTGu,Xπu,ϕudu=Vt,x,0.
According to (65) and (68), we get V^t,x,0=Vt,x,0. Finally, we apply (61) to ϕ∗,π∗ as the above process. Then,(69)V^t,x,0=Et,x,0Pϕ∗UXπ∗T+∫tTGu,Xπ∗u,ϕ∗udu=Vt,x,0.
Forh=1, we can obtain the following result by the similar method:(70)V^t,x,1=Et,x,1Pϕ∗UXπ∗T+∫tTGu,Xπ∗u,ϕ∗udu=Vt,x,1.
Thus,ϕ∗,π∗ is an optimal control strategy about (69)-(70), and V^t,x,h=Vt,x,h.Lemma 4.
The following integral is finite:(71)IT:=Eexp∫0Tα2β1ϕ1∗t2+α2β2ϕ2∗t2+α2β3ϕ3∗t2+ϕ4∗tlnϕ4∗t−ϕ4∗t+1β4hPα+λα∫−1∞ϕ5∗tφ∗t,ylnϕ5∗tφ∗t,y−ϕ5∗tφ∗t,y+1dF1yβ5dt<∞.Proof.
Puttingϕ∗ and π1∗ into (71), with an appropriate constant M1>0, we have(72)IT=Eexp∫0Tα2e2rT−tβ1σ2π1∗2+β2σ02+β3λ2h2a∗+ϕ4∗tlnϕ4∗t−ϕ4∗t+1β4hPα+λα∫−1∞ϕ5∗tφ∗t,ylnϕ5∗tφ∗t,y−ϕ5∗tφ∗t,y+1dF1yβ5dt≤M1Eexp∫0Tϕ4∗tlnϕ4∗t−ϕ4∗t+1β4hPα+λα∫−1∞ϕ5∗tφ∗t,ylnϕ5∗tφ∗t,y−ϕ5tφ∗t,y+1dF1yβ5dt<∞,where the inequality is established because π∗ and ϕ∗ are the deterministic and bound functions on 0,T.Lemma 5.
For the optimal control problem (21) with the exponential utility function (11), if V^t,x,h is the solution of (24) with the boundary condition V^T,x,h=Ux, then we have(73)EPϕ∗supt∈0,TV^t,Xπ∗t,Ht4<∞,EPϕ∗supt∈0,TGt,Xπ∗t,ϕ∗t2<∞.Proof.
Substitutingπ∗,ϕ∗ into (19), we have the wealth process under π∗,ϕ∗:(74)Xπ∗t=ertXπ∗0+∫0te−ru−tμ−rπ1∗u+π2∗u1−Huδ+λ2θ0−ημZ+h1a∗−ϕ1∗uπ1∗uσ−σ0ϕ2∗u−ϕ3∗uλ2h2a∗du+∫0te−ru−tπ1∗uσdW1Pϕ∗u+∫0te−ru−tσ0dW2Pϕ∗u+∫0te−ru−tλ2h2a∗dW3Pϕ∗u−∫0te−ru−tπ2∗u1−Hu−ζdHu+∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy.
For the candidate value function (21), we get(75)V^t,Xπ∗t,Ht4=1−HtV^t,Xπ∗t,0+HtV^t,Xπ∗t,14≤4V^t,Xπ∗t,04+4V^t,Xπ∗t,14.
At first, we proveEPϕ∗supt∈0,TV^t,Xπ∗t,04<∞. π∗,ϕ∗, and g2t are deterministic continuous functions and are bounded on 0,T. Substituting (75) into (48), there are two constants K1 and K2 satisfying 0<K1<K2 such that(76)V^t,Xπ∗t,04=g24tα4exp−4αertXπ∗0+∫0te−ru−tμ−rπ1∗u+π2∗u1−Huδ+λ2θ0−ημZ+h1a∗−ϕ1∗uπ1∗uσ−σ0ϕ2∗u−ϕ3∗uλ2h2a∗du+∫0te−ru−tπ1∗uσdW1Pϕ∗u+∫0te−ru−tσ0dW2Pϕ∗u+∫0te−ru−tλ2h2a∗dW3Pϕ∗u−∫0te−ru−tπ2∗u1−Hu−ζdHu+∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy≤K1exp−4α∫0te−ru−tπ1∗uσdW1Pϕ∗u+∫0te−ru−tσ0dW2Pϕ∗u+∫0te−ru−tλ2h2a∗dW3Pϕ∗u−∫0te−ru−tπ2∗u1−Hu−ζdHu+∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy=K1exp4α∫0te−ru−tπ2∗u1−Hu−ζdHu+32α2∫0te−2ru−tπ1∗2uσ2+σ02+λ2h2a∗du−32α2∫0te−2ru−tπ1∗2uσ2du−4α∫0te−ru−tπ1∗uσdW1Pϕ∗u−32α2∫0te−2ru−tσ02du−4α∫0te−ru−tσ0dW2Pϕ∗u−32α2∫0te−2ru−tλ2h2a∗du−4α∫0te−ru−tλ2h2a∗dW3Pϕ∗u−4α∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy≤K2exp−32α2∫0te−2ru−tπ1∗2uσ2du−4α∫0te−ru−tπ1∗uσdW1Pϕ∗u−32α2∫0te−2ru−tσ02du−4α∫0te−ru−tσ0dW2Pϕ∗u−32α2∫0te−2ru−tλ2h2a∗du−4α∫0te−ru−tλ2h2a∗dW3Pϕ∗u−4α∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy≜K2expD1t+D2t+D3t+D4t,where(77)D1t=−32α2∫0te−2ru−tπ1∗2uσ2du−4α∫0te−ru−tπ1∗uσdW1Pϕ∗u,D2t=−32α2∫0te−2ru−tσ02du−4α∫0te−ru−tσ0dW2Pϕ∗u,D3t=−32α2∫0te−2ru−tλ2h2a∗du−4α∫0te−ru−tλ2h2a∗dW3Pϕ∗u,D4t=−4α∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy,where π∗andϕ∗ are deterministic continuous functions, which are bounded on 0,T, and we get(78)EPϕ∗exp4D4t<EPϕ∗exp−16α∫0t∫−1∞e−ru−tπ1∗uyN˜Pϕ∗du,dy<∞.
Applying Lemma5 in Zeng and Taksar [43], we know that exp4D1t, exp4D2t, and exp4D3t are martingales; then,(79)EPϕ∗exp4D1t<∞,(80)EPϕ∗exp4D2t<∞,(81)EPϕ∗exp4D3t<∞.
Applying (78)–(81) with an appropriate constant K3 satisfying K3>K2, we have(82)EPϕ∗V^t,Xπ∗t,04≤K3EPϕ∗expD1t+D2t+D3t+D4t≤K3EPϕ∗exp2D3t+2D4tEPϕ∗exp2D1t+2D2t1/2≤K3EPϕ∗exp4D3tEPϕ∗exp4D4t×EPϕ∗exp4D1tEPϕ∗exp4D2t1/4<∞.
Consequently,EPϕ∗supt∈0,TV^t,Xπ∗t,04<∞. Similarity, we can also prove(83)EPϕ∗supt∈0,TV^t,Xπ∗t,14<∞.
Then, the formulaEPϕ∗supt∈0,TV^t,Xπ∗t,Ht4<∞ holds. The first part of Lemma 4 is proved.
Let(84)ft=α2β1ϕ1∗t2+α2β2ϕ2∗t2+α2β3ϕ3∗t2+ϕ4∗tlnϕ4∗t−ϕ4∗t+1β4hPα+λα∫−1∞ϕ5∗tφ∗t,ylnϕ5∗tφ∗t,y−ϕ5∗tφ∗t,y+1dF1yβ5,where ft is obviously bounded, then we get(85)EPϕ∗supt∈0,TGt,Xπ∗t,ϕ∗t2=EPϕ∗supt∈0,Tft2V^t,Xπ∗t,h2≤EPϕ∗supt∈0,Tft41/2EPϕ∗supt∈0,TH^t,Xπ∗t,h41/2<∞.
The first inequality follows from Cauchy–Schwarz inequality. The last inequality follows from (71). Lemma 5 is proved.Based on the discussion above, the main theorem is summarized as follows.Theorem 3.
For the robust optimal control problem (21) with the exponential utility function (11), V^t,x,h is the solution of (24) with the boundary condition V^T,x,h=Ux, π∗,ϕ∗ is an optimal strategy, and then Vt,x,h=V^t,x,h is the corresponding value function.Proof.
From Lemmas1, 2, and 3 and Theorems 1 and 2, we can obtain properties 1–4 in Proposition 1. By Lemma 5, condition 5 in Proposition 1 also holds for V^t,x,h. From Proposition 1, we can obtain the result of Theorem 3, π∗,ϕ∗ is an optimal strategy, and Vt,x,h is the corresponding value function.


## 5. Some Special Cases

In this section, we shall present some special cases of our results, such as the proportional reinsurance, the excess-of-loss reinsurance, and an ambiguity-neutral insurer of the insurance.


### 5.1. The Proportional Reinsurance

The parameterη satisfies η=0, and the insurer purchases reinsurance in the form of the proportional reinsurance, that is, ℋa,Z=2aθ1Z. The result is shown by corollary as follows.Corollary 5.1.
Ifη=0, then the value function Vt,x,h=1−hVt,x,0+hVt,x,1,h=0,1, and the optimal strategies π∗ and ϕ∗ are given by(86)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=2β3λ2a∗tθ1σZerT−t,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,t∈0,T,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,τ∧T,0,t∈τ∧T,T,a∗t=12θ1+α+β3erT−t,t∈0,T.The optimal value functionVt,x,h is as follows:(87)Vt,x,1=−1αexp−αxerT−tg1t,andVt,x,0=−1αexp−αxerT−tg2t,where(88)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−u−θ11−2a∗uθ12σZ2−2α+β3a∗u2θ12σZ2erT−udu,and g2t=eg¯2t, in which(89)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−u−θ11−2a∗uθ12σZ2−2α+β3a∗u2θ12σZ2erT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Proof.
In this case,ℋa,Z=2aθ1Z, and then the functions in our results are as follows:(90)h1a=ηh3a−1+ηθ1h4a=−θ11−2aθ12σZ2,h2a=Eℋa,Z2=4a2θ12σZ2,h3a=Eℋa,Z=2aθ1μZ,h4a=EZ−ℋa,Z2=1−2a∗θ12σZ2.
According to Theorems 3.3 and 3.5, we can obtain the result.Remark 2.
Ifη=0, and 2aθ1=q, then ℋZi=2aθ1Zi=qZi becomes a proportional reinsurance type. There are similar studies, for example, Zhou et al. [24], Zheng et al. [26], and Wang et al. [44].


### 5.2. The Excess-of-Loss Reinsurance

The parameterθ1 satisfies θ1=0, and the insurer purchases reinsurance in the form of the excess-of-loss reinsurance, that is, ℋa,Z=aη∧Z. The result is shown by corollary as follows.Corollary 5.3.
Ifθ1=0, then the optimal value function Vt,x,h=1−hVt,x,0+hVt,x,1,h=0,1, and the optimal strategies π∗ and ϕ∗ are given by(91)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=β3λ2h2aerT−t,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,t∈0,T,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,τ∧T,0,t∈τ∧T,T,a∗t=1α+β3erT−t,t∈0,TThe optimal value functionVt,x,h is as follows:(92)Vt,x,1=−1αexp−αxerT−tg1t,Vt,x,0=−1αexp−αxerT−tg2t,where(93)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−udu,and g2t=eg¯2t, in which(94)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Proof.
In this case,ℋa,Z=aη∧Z, and then the functions in our results are as follows:(95)h1a=ηh3a−1+ηθ1h4a=η∫0aη1−F2zdz,h2a=Eℋa,Z2=2∫0aη1−F2zdz,h3a=Eℋa,Z=∫0aη1−F2zdz.
According to Theorems1 and 2, we can obtain the result.Remark 3.
Ifθ1=0, then ℋZi=aη∧Zi becomes an excess-of-loss reinsurance type. Without regard for the default bond, let η=1, then the reinsurance strategy becomes the result of Li et al. [10].


### 5.3. Ambiguity-Neutral Insurer (ANI) Case

If the insurer is an ambiguity-neutral insurer, then the aversion ambiguity coefficientβi=0,i=1,2,3,4,5. In this case, for an admissible control aˇt,πˇ1t,πˇ2t and an initial value x,h, the objective function is described by(96)Vˇt,x,h=supπ∈ΠEUXπˇTXt,Ht=x,h.The corresponding HJB equation is(97)0=supπ∈ΠVˇt+Vˇxrx+μ−rπ1t+1−hδπ2t+λ2θ0−ημZ+h1at+12Vxxσ2π12t+σ02+λ2h2at+λ1EPϕVˇt,x+π1tY,h−Vˇt,x,h+hP1−hVˇt,x−π2tζ,1−Vˇt,x,0.From the value functions(98)Vt,x,1=−1αexp−αxerT−tg10t,Vt,x,0=−1αexp−αxerT−tg20t,we can get the following candidate optimal strategies by the same way:(99)π2∗ˇt=e−rT−tσ2αμ−r+λ1Ee−απ2∗ˇterT−tY,t∈0,T,π2∗ˇt=1αζe−rT−tlng20tΔg10tt∈0,τ∧T,0,t∈τ∧T,T,a∗ˇt=121+ηθ1+αerT−t,t∈0,T.where(100)g10t=exp∫tTα22e2rT−uσ2π1∗u2+σ02+λ2h2a∗u−αerT−uμ−rπ1∗u+λ2θ0−ημZ+h1a∗u+λ1Ee−απ1∗uerT−uY−1du,and g20t=eg¯20t, in which(101)g¯20t=e−δ/ζT−t∫tTeδ/ζT−uα22e2rT−uσ2π1∗u2+σ02+λ2h2a∗u+δζΔ−1−1δ−αerT−uμ−rπ1∗u+λ2θ0−ημZ+h1a∗u+λ1Ee−απ1∗uerT−uY−1du.Remark 4.
If all of the ambiguity aversion coefficients equal 0, i.e.,βi=0,i=1,2,3,4,5, our model reduces to an optimization problem for an ambiguity-neutral insurer (ANI). For the ANI, the optimization investment-reinsurance is researched by Cao [11], Yang and Zhang [2], etc.


## 5.1. The Proportional Reinsurance

The parameterη satisfies η=0, and the insurer purchases reinsurance in the form of the proportional reinsurance, that is, ℋa,Z=2aθ1Z. The result is shown by corollary as follows.Corollary 5.1.
Ifη=0, then the value function Vt,x,h=1−hVt,x,0+hVt,x,1,h=0,1, and the optimal strategies π∗ and ϕ∗ are given by(86)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=2β3λ2a∗tθ1σZerT−t,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,t∈0,T,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,τ∧T,0,t∈τ∧T,T,a∗t=12θ1+α+β3erT−t,t∈0,T.The optimal value functionVt,x,h is as follows:(87)Vt,x,1=−1αexp−αxerT−tg1t,andVt,x,0=−1αexp−αxerT−tg2t,where(88)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−u−θ11−2a∗uθ12σZ2−2α+β3a∗u2θ12σZ2erT−udu,and g2t=eg¯2t, in which(89)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−u−θ11−2a∗uθ12σZ2−2α+β3a∗u2θ12σZ2erT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Proof.
In this case,ℋa,Z=2aθ1Z, and then the functions in our results are as follows:(90)h1a=ηh3a−1+ηθ1h4a=−θ11−2aθ12σZ2,h2a=Eℋa,Z2=4a2θ12σZ2,h3a=Eℋa,Z=2aθ1μZ,h4a=EZ−ℋa,Z2=1−2a∗θ12σZ2.
According to Theorems 3.3 and 3.5, we can obtain the result.Remark 2.
Ifη=0, and 2aθ1=q, then ℋZi=2aθ1Zi=qZi becomes a proportional reinsurance type. There are similar studies, for example, Zhou et al. [24], Zheng et al. [26], and Wang et al. [44].


## 5.2. The Excess-of-Loss Reinsurance

The parameterθ1 satisfies θ1=0, and the insurer purchases reinsurance in the form of the excess-of-loss reinsurance, that is, ℋa,Z=aη∧Z. The result is shown by corollary as follows.Corollary 5.3.
Ifθ1=0, then the optimal value function Vt,x,h=1−hVt,x,0+hVt,x,1,h=0,1, and the optimal strategies π∗ and ϕ∗ are given by(91)ϕ1∗t=β1σπ1terT−t,t∈0,T,ϕ2∗t=β2σ0erT−t,t∈0,T,ϕ3∗t=β3λ2h2aerT−t,0=αhPβ4ϕ4∗tlnϕ4∗t+hPϕ4∗t−δζ,t∈0,T,φ∗t,y=1ϕ5∗texpβ5αe−απ1terT−ty−1,t∈0,T,ϕ5∗t=Eexpβ5αe−απ1terT−tY−1,t∈0,T,π1∗t=e−rT−tσ2α+β1μ−r+λ1∫−1∞ye−απ1∗terT−tyexpβ5αe−απ1∗terT−ty−1dF1y,t∈0,T,π2∗t=1αζe−rT−tlng2tΔϕ4∗tg1t,t∈0,τ∧T,0,t∈τ∧T,T,a∗t=1α+β3erT−t,t∈0,TThe optimal value functionVt,x,h is as follows:(92)Vt,x,1=−1αexp−αxerT−tg1t,Vt,x,0=−1αexp−αxerT−tg2t,where(93)g1t=exp∫tTα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−λ1αβ5−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−udu,and g2t=eg¯2t, in which(94)g¯2t=e−δ/ζT−t∫tTeδ/ζT−uα+β22ασ02e2rT−u−αλ2erT−uθ0−ημZ−αerT−uμ−rπ1∗u+α+β12ασ2e2rT−uπ1∗u2−λ1αβ5+λ1αβ5∫−1∞expβ5αe−απ1∗uerT−uy−1dF1y−αλ2erT−uh1a∗u−α+β32h2a∗uerT−u+δζlnΔϕ4∗u+lng1u+ϕ4∗u−1β4αΔdu.Proof.
In this case,ℋa,Z=aη∧Z, and then the functions in our results are as follows:(95)h1a=ηh3a−1+ηθ1h4a=η∫0aη1−F2zdz,h2a=Eℋa,Z2=2∫0aη1−F2zdz,h3a=Eℋa,Z=∫0aη1−F2zdz.
According to Theorems1 and 2, we can obtain the result.Remark 3.
Ifθ1=0, then ℋZi=aη∧Zi becomes an excess-of-loss reinsurance type. Without regard for the default bond, let η=1, then the reinsurance strategy becomes the result of Li et al. [10].


## 5.3. Ambiguity-Neutral Insurer (ANI) Case

If the insurer is an ambiguity-neutral insurer, then the aversion ambiguity coefficientβi=0,i=1,2,3,4,5. In this case, for an admissible control aˇt,πˇ1t,πˇ2t and an initial value x,h, the objective function is described by(96)Vˇt,x,h=supπ∈ΠEUXπˇTXt,Ht=x,h.The corresponding HJB equation is(97)0=supπ∈ΠVˇt+Vˇxrx+μ−rπ1t+1−hδπ2t+λ2θ0−ημZ+h1at+12Vxxσ2π12t+σ02+λ2h2at+λ1EPϕVˇt,x+π1tY,h−Vˇt,x,h+hP1−hVˇt,x−π2tζ,1−Vˇt,x,0.From the value functions(98)Vt,x,1=−1αexp−αxerT−tg10t,Vt,x,0=−1αexp−αxerT−tg20t,we can get the following candidate optimal strategies by the same way:(99)π2∗ˇt=e−rT−tσ2αμ−r+λ1Ee−απ2∗ˇterT−tY,t∈0,T,π2∗ˇt=1αζe−rT−tlng20tΔg10tt∈0,τ∧T,0,t∈τ∧T,T,a∗ˇt=121+ηθ1+αerT−t,t∈0,T.where(100)g10t=exp∫tTα22e2rT−uσ2π1∗u2+σ02+λ2h2a∗u−αerT−uμ−rπ1∗u+λ2θ0−ημZ+h1a∗u+λ1Ee−απ1∗uerT−uY−1du,and g20t=eg¯20t, in which(101)g¯20t=e−δ/ζT−t∫tTeδ/ζT−uα22e2rT−uσ2π1∗u2+σ02+λ2h2a∗u+δζΔ−1−1δ−αerT−uμ−rπ1∗u+λ2θ0−ημZ+h1a∗u+λ1Ee−απ1∗uerT−uY−1du.Remark 4.
If all of the ambiguity aversion coefficients equal 0, i.e.,βi=0,i=1,2,3,4,5, our model reduces to an optimization problem for an ambiguity-neutral insurer (ANI). For the ANI, the optimization investment-reinsurance is researched by Cao [11], Yang and Zhang [2], etc.


## 6. Sensitivity Analysis

In this section, we will give several numerical examples to illustrate the influences of the parameters on the optimal strategies and the optimal value functions. Unless otherwise stated, the basic parameters are given in Table1.Table 1
Model parameters.
μ
T
r
α
σ
σ
0
λ
1
λ
2
θ
1
θ
0
δ
0.1
1
0.05
1
0.15
0.5
2
2
2
1
0.2Some analyses of the optimal reinsurance strategya∗t are shown in Figures 1–4. β3 is the ambiguity aversion coefficient of the AAI. From Figure 1, it is found that β3 affects the reinsurance strategy of the insurer. As β3 increases, the insurer has lower risk exposure in the insurance market, so less amount of money will be paid to purchase reinsurance. η and θ1 are the relative safety loadings of the reinsurer. Figures 2 and 3 show the effects of η or θ1 on the insurer’s reinsurance strategy. As η or θ1 increases, the reinsurer pays more concern on his/her risk exposures and charges more for them. Consequently, the insurer decreases his/her demand for reinsurance and pays more claims by himself/herself. In Figure 4, we show the common effects of the safety loading θ1 and the ambiguity aversion coefficient β3 on a∗t.Figure 1
Effect ofβ3 on a∗t.Figure 2
Effect ofη on a∗t.Figure 3
Effect ofθ1 on a∗t.Figure 4
Effect ofθ1 and β3 on a∗t.In Figures5–8, we illustrate the impacts of the ambiguity aversion coefficients β1 and β5 on the stock strategy π1∗t. From Figures 5 and 6, we find that the AAI will reduce the wealth invested on the stock, when there is a higher ambiguity aversion coefficient. From Figures 7 and 8, we find that as the time t increases, the AAI increases the stock investment amount. These figures show that the robust optimal strategies can effectively reduce the sensitivity of π1∗t on the stock.Figure 5
Effect ofβ1 on π1∗t.Figure 6
Effect ofβ5 on π1∗t.Figure 7
Effect ofβ1 and the time t on π1∗t.Figure 8
Effect ofβ5 and the time t on π1∗t.For the defaultable corporate bond, the value is assumed to be zero after default. In Figures9–13, we show the numerical analysis of the defaultable corporate bond strategy π2∗t before default. β2 and β4 are the ambiguity aversion coefficients. As β4 increases, the insurer will reduce the money on the defaultable corporate bond, but β2 does not affect the investment, as shown in Figure 9. In Figure 10, the insurer will invest more amount of his/her money, if the defaultable corporate bond with a higher premium induces a higher potential yield. Contrary to the default risk premium 1/Δ, the accession of ζ reduces the insurer’s investment on the defaultable bond, as shown in Figure 11. A higher loss rate ζ leads to a less recovery value, which implies a higher potential loss of the insurer. When η=0, the reinsurance type is a proportional reinsurance, and when θ1=0, the reinsurance type becomes an excess-of-loss reinsurance. We show that the proportion reinsurance is always below the excess-of-loss reinsurance at the same time in Figure 12. Figure 13 provides a full description of π2∗t with respect to 1/Δ and ζ with two different reinsurance types, respectively.Figure 9
Effect ofβ2 and β4 on π2∗t.Figure 10
Effect of1/Δ on π2∗t.Figure 11
Effect ofζ on π2∗t.Figure 12
Effect ofθ1=0 and η=0 on π2∗t.Figure 13
Effect of1/Δ and ζ on π2∗t with θ1=0 and η=0.
(a)
(b)In Figure14, we illustrate the predefault value function V0,x,0 and the postdefault value function V0,x,1 with respect to the initial wealth about a proportional reinsurance and an excess-of-loss reinsurance, respectively. We can see that the value functions of the insurer increase as the initial wealth increases and the predefault value function is always greater than or equal to the postdefault value function.Figure 14
Value functions with respect tox when θ1=0 and η=0.
(a)
(b)


## 7. Conclusion

In this paper, we consider a robust optimal reinsurance-investment problem of an insurer under the generalized mean-variance premium principle and a defaultable market. The insurer can trade in a risk-free asset, a stock, and a defaultable corporate bond. The surplus of the AAI is described by an approximate diffusion process. The stock’s price process is described by a jump-diffusion model. Using the dynamic programming approach, we study the predefault case and the postdefault case, respectively, and derive the optimal strategies and the corresponding value functions under the worst-case scenario. We give some sensitivity analysis to illustrate our theoretical results. In future research, we will consider some complex models, such as the robust optimal reinsurance-investment problem of stochastic differential games.



---

*Source: 6207805-2020-03-11.xml*