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

LCSM, Ministry of Education, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China School of Mathematics and Physics, Nanyang Institute of Technology, Nanyang, Henan 473004, China Key Laboratory of Applied Statistics and Data Science, Hunan Normal University, College of Hunan Province, Changsha, Hunan 410081, China School of Business, Hunan Normal University, Changsha, Hunan 410081, China


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. e insurer can also transfer their risks by buying reinsurance. erefore, 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]. is is because in the face of serious events (natural disasters and serious large-scale diseases), the stock price may jump to a new level. erefore, 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. erefore, 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. e 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 investmentinsurance 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. e 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. erefore, 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 P and the alternative measure P ϕ , but the jump intensity is uncertain.
is 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. erefore, 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]. e 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 closedform 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.

Model Formulation
In this article, we consider a complete probability space (Ω, F, P). Let F: � (F t ) t≥0 be the right-continuous, P-complete filtration generated by two standard Brownian motions W 1 (t) and W 2 (t) , a Poisson process N(t) { }, and two families of random variables Y i , i ≥ 1 and Z i , i ≥ 1 . We assume that W 1 (t) , W 2 (t) , N(t) { }, Y i , and Z i are mutually independent. Let G: � (G t ) t≥0 be the enlarged filtration of F and H, i.e., G t : We assume that every F-martingale is also a G-martingale. e probability measure P is the real-world probability measure, and Q is the risk-neutral measure.

e Financial Market.
In this section, we consider a financial market consisting of three types of securities: a riskfree asset, a stock, and a defaultable corporate bond. e price process of the risk-free asset under the measure P is described by where r > 0 is the risk-free interest rate. e price process S(t) { } t≥0 of a stock is described by a jump-diffusion process: where μ > 0 is the expected instantaneous rate of return of the stock; σ is a positive constant; We assume that P Y i > − 1 forall i ≥ 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 + λ 1 μ Y > r.
Next, we consider that N(dt, dy) is a Poisson random measure on Ω × [0, T] × (− 1, ∞) and υ(dt, dy) is the compensator measure of N(dt, dy). at is, N(dt, dy) � N(dt, dy) − υ(dt, dy), and then where υ(dt, dy) � λ 1 dt dF 1 (y). By the definition of F, for any A ∈ B((− 1, ∞)), 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 h P > 0 under P. A default indicator process is defined as H(t): � I τ≤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 H(t) 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 M P (t) � H(t)− t 0 (1 − H(u))h P du, 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 h Q � h P /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 T 1 , 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]) dp t, T 1 � p t− , where δ : � ζh Q represents the risk-neutral credit spread and 0 < ζ < 1 denotes the loss rate of the bond when a default occurs.

Dynamics of Surplus
Process. e insurer's surplus process U 0 (t) t≥0 is described by a jump-diffusion risk model: where c is the premium rate and σ 0 ≥ 0 is a constant.
i�1 Z i represents the aggregate claim amount up to time t, where N 2 (t) is a homogeneous Poisson process with intensity λ 2 > 0, and the individual claim sizes Z 1 , Z 2 , . . . , independent of N 2 (t) , are i.i.d. positive random variables with the common distribution function F 2 (z), the first moment E[Z i ] � μ Z , and the second moment In addition, the insurance premium rate c under the expected value principle is given by c � λ 2 μ Z (1 + θ 0 ), where θ 0 > 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 Z i at time t, a proportion H(Z i ) (self-reinsurance function, see Schmidli [39]) is paid by the insurer, and the rest Z i − H(Z i ) is paid by the reinsurer. c H 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 where η ≥ 0 and θ 1 ≥ 0 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 H(·) may take various forms, Zhang et al. [5] supposed that where a ∈ [0, (1/2(1 + η)θ 1 )]. Next, we will only consider reinsurance strategies given by (8). In this case, since H(Z i ) is uniquely characterized by the parameter a, we also rewrite H(Z i ) as H(a, Z i ) to emphasize the dependence on a and call a as the insurer's reinsurance strategy. at is, H(a, Z i ) � a(η + 2(1 + η)θ 1 Z i ) ∧ Z i . 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 � 1/2(1 + η)θ 1 , H(a, Z i ) � Z i , 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]. en, the surplus process of the insurer under the retention H at time t is given by Mathematical Problems in Engineering 3 Remark 1. If η � 0, then H(Z i ) � 2aθ 1 Z i becomes a proportional reinsurance type (see also Zhou et al. [24] and Zheng et al. [26]); if θ 1 � 0, then H(Z i ) � (aη) ∧ Z i 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]). erefore, proportional reinsurance and excess-of-loss reinsurance are special cases of (8).

e Wealth Process.
In this section, we assume that the insurer is allowed to invest all his/her surplus in the financial market defined above. e insurer's trading strategy is π(t) � (π 1 (t), π 2 (t), a), where π 1 (t) is the total amount of wealth invested in the risky asset (a stock) at time t, π 2 (t) is the amount of wealth invested in the defaultable corporate bond, and a is the insurer's reinsurance strategy. e 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 < T 1 . e reserve process subjected to this choice is denoted by X π (T). us, the wealth process can be presented as follows: Suppose that the insurer has an exponential utility function defined by For an admissible control π(t) and an initial value (x, h), we define the objective function as We denote the set of all admissible strategies by Π. en, we have the following definition for the set of admissible strategies.
where P ϕ 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 probability P, 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 : � P ϕ | P ϕ ∼ P equivalent to P (Anderson et al. [27] and Zeng et al. [6]). At first, we define a process ϕ(t, y) and φ(t, 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 P ϕ ∈ P has a Radon-Nikodym derivative: with respect to P, where the process Λ ϕ (t) is modelled by the stochastic differential equation (see Jin et al. [38]) 4 Mathematical Problems in Engineering with Λ ϕ (0) � 1, P-a.s. By the Itô differentiation rule, we get Note that ϕ 5 (t) and φ(t, y) are positive stochastic processes, and φ(t, y) satisfies the following relationship: e distribution of the stock's jump amplitude and the jump intensity is ambiguous, so the density function is not equal under P and P ϕ . Under the probability measure P ϕ , the jump intensity λ 1 and the density function dF 1 (y) of the stock's price process are changed into ϕ 5 (t)λ 1 and φ(t, y)dF 1 (y) in the alternative model.
{ } , t ≥ 0, the intensity h P of the jumps becomes ϕ 4 (t)h P , and the jump size is always equal to 1, so the jump size distribution is identical under P and P ϕ . For three standard Brownian motions, according to Girsanov's theorem, To simplify further analysis, we define the following functions: Mathematical Problems in Engineering en, the dynamics of the wealth process under P ϕ is 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]. e insurer penalizes any deviation from this reference model and the penalty increases with this deviation. en, we use relative entropy to measure the deviation between the reference measure P and an alternative measure P ϕ . e increase in relative entropy from t to t + dt is shown by with h ∈ 0, 1 { }. e 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 where E P ϕ x,h is calculated under the alternative measure P ϕ , the initial values of the processes are given by e five terms in (22) are scaled by Ψ 1 ≥ 0, Ψ 2 ≥ 0, Ψ 3 ≥ 0, Ψ 4 ≥ 0, and Ψ 5 ≥ 0, which are state-dependent. We follow Maenhout [29] and set , , , , where β i ≥ 0, i � 1, 2, 3, 4, 5, is the ambiguity aversion coefficient with respect to three diffusion risks and two jump risks. e larger the values of Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 , and Ψ 5 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.

The Main Result
In this section, the goal is to find the optimal allocation pair (π 1 (t), π 2 (t), a(t)) under the worst-case scenario. According to the dynamic programming principle, the HJB equation can be derived as (Anderson et al. [42]) 6 Mathematical Problems in Engineering where A π,ϕ is the infinitesimal generator of (21) under P ϕ and is defined by where V t , V x , and V xx 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: 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.

e Postdefault Case.
In this section, we will concentrate on the postdefault case, that is, HJB equation (27) for V(t, x, 1), and we conjecture that the value function has the following form: Mathematical Problems in Engineering where g 1 (t) is a deterministic function, with g 1 (T) � 1. We get Substituting these partial derivatives and E[·] 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 E[φ(t, Y)] � 1 (formula (16)), we have Lemma 1. For any t ∈ [0, T], the equation has a unique positive solution π * 1 (t).

Mathematical Problems in Engineering
Proof. Suppose which implies that W(t, π 1 ) is a decreasing function w.r.t. π 1 .
Theorem 1 (postdefault strategy). e robust optimal reinsurance and investment strategies for the period after default are given as Furthermore, the postdefault value function is given by where
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 π * ′ 1 � π * 1 . Substituting π * ′ and ϕ * ′ into (44), we get the equation Note that g 2 (t) > 0; in order to get the expression for g 2 (t) with the boundary condition g 2 (T) � 1, we try the following form of g 2 (t) � e g 2 (t) : erefore, we can derive the following theorem.
□ Theorem 2 (predefault strategy). e robust optimal reinsurance and investment strategies for the period before default are given as Furthermore, the predefault value function is given by where g 2 (t) � e g 2 (t) , in which Next, putting the predefault and postdefault cases together, we have the following solution to the HJB equation (24) associated with the value function V(t, x, h): (t, x, 1), where h � 0 or 1.
In the next section, we shall show that the above stochastic control policies are indeed the optimal strategies and that the value function V(t, x, h) is unique.

Verification Theorem
In order to verify the candidate optimal strategies π * and ϕ * are indeed optimal, and the candidate value function is (55), we give the verification theorem as follows.  H(τ)) τ∈T and G(τ, X π (τ), ϕ(τ)) τ∈T are uniformly integrable, where T denotes the set of stopping times τ ≤ T , h) and (ϕ * , π * ) are an optimal control.

Lemma 4.
e following integral is finite: Proof. Putting ϕ * and π * 1 into (71), with an appropriate constant M 1 > 0, we have where the inequality is established because π * and ϕ * are the deterministic and bound functions on [0, T].

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Mathematical Problems in Engineering where π * and ϕ * are deterministic continuous functions, which are bounded on [0, T], and we get Applying Lemma 5 in Zeng and Taksar [43], we know that exp 4D 1 (t) , exp 4D 2 (t) , and exp 4D 3 (t) are martingales; then, Applying (78)-(81) with an appropriate constant K 3 satisfying K 3 > K 2 , we have Consequently, en, the formula where f(t) is obviously bounded, then we get e first inequality follows from Cauchy-Schwarz inequality. e 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) � U(x), (π * , ϕ * ) is an optimal strategy, and then V(t, x, h) � V(t, x, h) is the corresponding value function.
Proof. From Lemmas 1, 2, and 3 and eorems 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 eorem 3, (π * , ϕ * ) is an optimal strategy, and V(t, x, h) is the corresponding value function.

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.

e Proportional Reinsurance.
e parameter η satisfies η � 0, and the insurer purchases reinsurance in the form of the proportional reinsurance, that is, H(a, Z) � 2aθ 1 Z. e result is shown by corollary as follows. (t, x, 1), h � 0, 1, and the optimal strategies π * and ϕ * are given by (86) e optimal value function V(t, x, h) is as follows: where e optimal value function V(t, x, h) is as follows: where and g 2 (t) � e g 2 (t) , in which Proof. In this case, H(a, Z) � aη ∧ Z, and then the functions in our results are as follows: According to eorems 1 and 2, we can obtain the result.

Ambiguity-Neutral Insurer (ANI) Case.
If the insurer is an ambiguity-neutral insurer, then the aversion ambiguity coefficient β i � 0, i � 1, 2, 3, 4, 5. In this case, for an admissible control (� a(t), � π 1 (t), � π 2 (t)) and an initial value (x, h), the objective function is described by e corresponding HJB equation is From the value functions we can get the following candidate optimal strategies by the same way: where and g 20 (t) � e g 20 (t) , in which Remark 4. If all of the ambiguity aversion coefficients equal 0, i.e., β i � 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.

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 Table 1.
Some analyses of the optimal reinsurance strategy a * (t) are shown in Figures 1-4. β 3 is the ambiguity aversion coefficient of the AAI. From Figure 1, it is found that β 3 affects the reinsurance strategy of the insurer. As β 3 increases, the insurer has lower risk exposure in the insurance market, so less amount of money will be paid to purchase reinsurance. η and θ 1 are the relative safety loadings of the

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Mathematical Problems in Engineering reinsurer. Figures 2 and 3 show the effects of η or θ 1 on the insurer's reinsurance strategy. As η or θ 1 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 θ 1 and the ambiguity aversion coefficient β 3 on a * (t).
In Figures 5-8, we illustrate the impacts of the ambiguity aversion coefficients β 1 and β 5 on the stock strategy π * 1 (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. ese figures show that the robust optimal strategies can effectively reduce the sensitivity of π * 1 (t) on the stock. For the defaultable corporate bond, the value is assumed to be zero after default. In Figures 9-13, we show the numerical analysis of the defaultable corporate bond strategy π * 2 (t) before default. β 2 and β 4 are the ambiguity aversion coefficients. As β 4 increases, the insurer will reduce the money on the defaultable corporate bond, but β 2 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 θ 1 � 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 π * 2 (t) with respect to 1/Δ and ζ with two different reinsurance types, respectively.
In Figure 14, we illustrate the predefault value function V(0, x, 0) and the postdefault value function V(0, 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.

Conclusion
In this paper, we consider a robust optimal reinsuranceinvestment problem of an insurer under the generalized mean-variance premium principle and a defaultable market. e insurer can trade in a risk-free asset, a stock, and a defaultable corporate bond. e surplus of the AAI is described by an approximate diffusion process. e stock's price process is described by a jump-diffusion model. Using

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Mathematical Problems in Engineering 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.

Data Availability
All data generated or analyzed during this study are included in this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.