Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2020 / Article
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Stochastic Process Theory and Its Applications

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Research Article | Open Access

Volume 2020 |Article ID 6207805 |

Man Li, Yingchun Deng, Ya Huang, Hui Ou, "Optimal Strategies for an Ambiguity-Averse Insurer under a Jump-Diffusion Model and Defaultable Risk", Mathematical Problems in Engineering, vol. 2020, Article ID 6207805, 26 pages, 2020.

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

Academic Editor: Wenguang Yu
Received24 Dec 2019
Accepted04 Feb 2020
Published11 Mar 2020


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.

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 measure and the alternative measure , 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: Section 2 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 space . Let be the right-continuous, -complete filtration generated by two standard Brownian motions and , a Poisson process , and two families of random variables and . We assume that , , and are mutually independent. Let be the enlarged filtration of and , i.e., , where the filtration is generated by a default process . We assume that every -martingale is also a -martingale. The probability measure 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 measure is described bywhere is the risk-free interest rate. The price process of a stock is described by a jump-diffusion process:where is the expected instantaneous rate of return of the stock; σ is a positive constant; is a homogeneous Poisson process with intensity , representing the number of a stock price’s jumps during the time interval ; and is the ith jump amplitude of the stock’s price, and are i.i.d. random variables with the common distribution function , the first moment , and the second moment . We assume that 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 .

Next, we consider that is a Poisson random measure on and is the compensator measure of . That is, , and thenwhere . By the definition of , for any , is a -martingale.

Next, we consider the price process of the defaultable corporate bond by the intensity-based approach. Let τ be the time of default and τ represent the first jump time of a Poisson process with constant intensity under P. A default indicator process is defined as for each , and the value of the corporate bond is assumed to be zero after default. Let be the filtration generated by the default process and augmented in the usual way. By definition, τ is naturally an -stopping time and a -stopping time. Furthermore, the martingale default process is thus given by , which is a -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 . We denote that is the default risk premium. We assume that there exists a defaultable zero coupon bond with a maturity date , 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])where represents the risk-neutral credit spread and denotes the loss rate of the bond when a default occurs.

2.2. Dynamics of Surplus Process

The insurer’s surplus process is described by a jump-diffusion risk model:where c is the premium rate and is a constant. represents the aggregate claim amount up to time t, where is a homogeneous Poisson process with intensity , and the individual claim sizes independent of , are i.i.d. positive random variables with the common distribution function , the first moment , and the second moment .

In addition, the insurance premium rate c under the expected value principle is given by , where 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 at time t, a proportion (self-reinsurance function, see Schmidli [39]) is paid by the insurer, and the rest is paid by the reinsurer. 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 iswhere and is the relative safety loading of the reinsurer.

According to Grandell [40], the surplus process can be approximated by the following diffusion process:

While the self-reinsurance function may take various forms, Zhang et al. [5] supposed thatwhere . Next, we will only consider reinsurance strategies given by (8). In this case, since is uniquely characterized by the parameter a, we also rewrite as to emphasize the dependence on a and call a as the insurer’s reinsurance strategy. That is, . 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 , , that is, the insurer pays all of the claims by himself/herself; when , 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 at time t is given by

Remark 1. If , then becomes a proportional reinsurance type (see also Zhou et al. [24] and Zheng et al. [26]); if , then 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 is , where is the total amount of wealth invested in the risky asset (a stock) at time t, 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 , where . The reserve process subjected to this choice is denoted by . Thus, the wealth process can be presented as follows:

Suppose that the insurer has an exponential utility function defined by

For an admissible control and an initial value , we define the objective function as

We denote the set of all admissible strategies by . Then, we have the following definition for the set of admissible strategies.

Definition 1. A trading strategy is said to be admissible if(i) is -progressively measurable(ii)(iii), the stochastic differential equation (10) has a pathwise unique solution with , where is the chosen measure to describe the worst case and will be shown laterHowever, the AAI wants to guard himself/herself against worst-case scenarios. We assume that the knowledge of the AAI about ambiguity is described by probability , 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 equivalent to (Anderson et al. [27] and Zeng et al. [6]). At first, we define a process such that(1) is -measurable, for each (2) and a.s. (3) and , P-a.s.The alternative measures , and are positive stochastic processes. We write for the space of all such processes ϕ. Note that P is the probability measure associated with the reference model. For every , each probability measure has a Radon–Nikodym derivative:with respect to P, where the process is modelled by the stochastic differential equation (see Jin et al. [38])with , P-a.s. By the Itô differentiation rule, we getNote that and are positive stochastic processes, and satisfies the following relationship:The distribution of the stock’s jump amplitude and the jump intensity is ambiguous, so the density function is not equal under P and . Under the probability measure , the jump intensity and the density function of the stock’s price process are changed into and in the alternative model. That is, , and is a compensated Poisson random measure. For the default indicator process the intensity of the jumps becomes , and the jump size is always equal to 1, so the jump size distribution is identical under P and . For three standard Brownian motions, according to Girsanov’s theorem, Thus, the wealth process under becomes thatTo simplify further analysis, we define the following functions:Then, the dynamics of the wealth process under isNext, 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 . The increase in relative entropy from t to is shown bywith . 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 aswhere is calculated under the alternative measure , the initial values of the processes are given by , andThe five terms in (22) are scaled by , , , , and , which are state-dependent. We follow Maenhout [29] and setwhere , is the ambiguity aversion coefficient with respect to three diffusion risks and two jump risks. The larger the values of 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 pair under the worst-case scenario. According to the dynamic programming principle, the HJB equation can be derived as (Anderson et al. [42])with the boundary condition , where is the infinitesimal generator of (21) under and is defined bywhere represent the value function’s partial derivatives with respect to the corresponding variables. We split equation (24) into two cases: the postdefault case () and the predefault case (), and denote V as follows:

According to (26) and (21), the HJB equation (24) transforms into the following two forms:

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 , and we conjecture that the value function has the following form:where is a deterministic function, with . We get

Substituting these partial derivatives and into equation (27), we get

Fixing π and a and maximizing over ϕ yield the following first-order condition for the minimum point (there is no ambiguity about the default jump risk after default):

Noting that (formula (16)), we have

Lemma 1. For any , the equation has a unique positive solution .

Proof. Suppose , then we getwhich implies that is a decreasing function w.r.t. . Furthermore, we have . Also, we can find that if , we have . Therefore, equation (34) has a unique positive root .

Lemma 2. For the functions , , defined above, we have the following relationship:

Substituting into (31), according to the first-order condition and Lemmas 1 and 2, we can obtain the maximum point given by

Substituting and into (31), we get the equation

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

Furthermore, the postdefault value function is given bywhere

3.2. The Predefault Case

In this section, we will concentrate on the predefault case, that is, HJB equation (28) for , and we conjecture that the value function has the following form:where is a deterministic function, with . We get

Substituting these partial derivatives, , and into equation (28), we get