数学杂志  2022, Vol. 42 Issue (4): 300-314   PDF    
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本文作者相关文章
MU Rui
MA Shi-xia
ZHANG Xin-ru
ROBUST OPTIMAL INVESTMENT AND REINSURANCE STRATEGIES WITH DELAY AND DEFAULT RISK UNDER DEPENDENT RISK MODEL
MU Rui, MA Shi-xia, ZHANG Xin-ru    
School of Science, Hebei University of Technology, Tianjin 300401, China
Abstract: In this paper, we consider the robust optimal investment-reinsurance strategies with delay and default risk under risk dependent model. We assume that the ambiguity-averse insurer's wealth process have two dependent classes of insurance business and the surplus can invest in a risk-free asset, a defaultable bond and a risky asset whose price process satisfies Heston model. Using dynamic programming principle, we establish the robust Hamilton-Jacobi-Bellman(HJB) equations for the post-default case and the pre-default case, respectively. Furthermore, we obtain the robust optimal investment and reinsurance strategies and the corresponding value functions by maximizing the expected exponential utility of the terminal wealth. Finally, we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies.
Keywords: robust optimal strategies     heston model     stochastic differential delay equation     dependent risk     default risk    
相依风险模型下具有延迟和违约风险的鲁棒最优投资和再保险策略
慕蕊, 马世霞, 张欣茹    
河北工业大学理学院, 天津 300401
摘要:本文研究了在风险相依模型下具有延迟和违约风险的鲁棒最优投资再保险策略. 假设模糊厌恶型保险人的财富过程有两类相依的保险业务并且余额可以投资于无风险资产、可违约债券和价格过程遵循Heston模型的风险资产. 利用动态规划原则, 我们分别建立了违约后和违约前的鲁棒HJB方程. 另外, 通过最大化终端财富的期望指数效用, 我们得到了最优投资和再保险策略以及相应的值函数. 最后, 通过一些数值例子说明了某些模型参数对鲁棒最优策略的影响.
关键词鲁棒最优策略    Heston模型    随机微分延迟方程    相依风险    违约风险    
1 Introduction

Reinsurance and investment are two important issues for insurers. Reinsurance is an effective way to reduce risk, while investment is the most common way to increase wealth. In recent years, based on different decision-making objectives, more attention have been paid to the research on the optimal reinsurance-investment problem. For example, Liang et al.[1] considered the objective function of minimizing the ruin probability; Guan and Liang[2] studied the optimal problems aiming to maximize the expected utility of terminal wealth; Wang et al.[3] investigated optimal reinsurance and investment problems under mean variance criterion, and so on.

Besides the different decision-making objectives mentioned above, many scholars also studied the optimal investment-reinsurance strategy in the following aspects. Firstly, the model uncertainties do exist widely in finance and insurance, some scholars have investigated the impact of ambiguity on optimal reinsurance and investment problems. For example, Chen and Yang[4] considered the robust optimal reinsurance-investment strategy selection problem for an ambiguity-averse insurer (AAI). Wang et al.[5] investigated a class of robust non-zero-sum reinsurance-investment stochastic differential games between two competing insurers. More often, Zhang and Chen[6] and Sun et al.[7] also took the ambiguity into account. Secondly, the default of one company usually has strong influence on other companies, high yield bonds with default risk have become attractive to investors. Many scholars have studied the reinsurance-investment problem with default risk, such as Zhang and Chen[6], Yang[8], and so on. Thirdly, due to the insurer's current wealth existing capital inflow/outflow, it is necessary to take delay into account. Refer to Shen and Zeng[9], the optimal investment-reinsurance strategy is obtained under delay risk. Moreover, based on the different risk asset models, A and Li[10] and Zhang and Chen[11] studied an optimal excess-of-loss reinsurance and investment strategies with delay. In addition, some insurance businesses are usually correlated in practice. For example, a traffic accident may cause property loss or medical claims or death claims, these insurance businesses will be correlated. Therefore, it is necessary to take dependent risk into account. Bi et al.[12] studied the optimal problems with two dependent classes of insurance business under the criterion of mean-variance. Furthermore, with the multiple dependent risk, Yuen et al.[13] and Sun et al.[7] derived the optimal strategies and value function under the different risk models.

Inspired by the above references, we consider the robust optimal investment and reinsurance strategies with delay and default risk under risk dependent model. Comparing with the existing literature, the main contributions of this paper include: (ⅰ) We consider delay, default, dependent risk and ambiguity aversion simultaneously in an optimal reinsurance-investment problem; (ⅱ) We obtain the robust optimal strategies and corresponding value function in the post-default and pre-default case under risk dependent model.

The remainder of this paper is organized as follows. In section 2, we describe the model and the necessary assumptions. In Section 3, we state the robust optimal control problem for an AAI. Section 4 derives the robust optimal investment-reinsurance strategies under the post-default case and the pre-default case respectively. In section 5, we present some numerical illustrations to analyze our theoretical results.

2 Model Assumption

We consider a filtered complete probability space $ (\Omega, F, \left\{F_{t}\right\}_{t \in[0, T]}, P) $, where $ T $ represents the terminal time and is a positive finite constant and $ F_{t} $ stands for the information of the market available up to time $ t $. Assume that all processes introduced below are well-defined and adapted processes in this space.

2.1 Surplus Process

Assume that the insurer's surplus process is described by the following Cramer-Lundberg model

$ \begin{align} \mathrm{d} X(t)=c \mathrm{\; d} t-\mathrm{d}(\sum\limits_{i=1}^{N_{1}(t)+N(t)} X_{i})-\mathrm{d}(\sum\limits_{i=1}^{N_{2}(t)+N(t)} Y_{i}), \end{align} $ (2.1)

where $ X_{i} $ is the ith claim size from the first class; $ \left\{X_{i}, i \geq 1\right\} $ are assumed to be independent and identically distributed positive random variables with common distribution function $ F_{X}(\cdot) $, finite first moment $ E\left(X_{i}\right)=\mu_{1}>0 $, and second moment $ E\left(X_{i}^{2}\right)=\nu_{1} $; Similarly, $ Y_{i} $ is the $ i $th claim size from the second class, $ E\left(Y_{i}\right)=\mu_{2}>0 $, and $ E\left(Y_{i}^{2}\right)=\nu_{2} $; $ \left\{N_{1}(t), t \geq 0\right\} $, $ \left\{N_{2}(t), t \geq 0\right\} $ and $ \{N(t), t \geq 0\}$ are three independent Poisson processes with positive intensity parameters $ \lambda_{1} $, $ \lambda_{2} $ and $ \lambda $, respectively. $ \left\{X_{i}, i \geq 1\right\}, \left\{Y_{i}, i \geq 1\right\}, \left\{N_{1}(t), t \geq 0\right\}, \left\{N_{2}(t), t \geq\right. 0\}$ and $ \{N(t), t \geq 0\} $ are mutually independent. Further, the insurer's premium rate is calculated according to the expected value principle, i.e., $ c=\left(1+\eta\right)\left[\left(\lambda_{1}+\lambda\right) \mu_{1}+\left(\lambda_{2}+\lambda\right) \mu_{2}\right] $, where $ \eta>0 $ is the insurer's safety loading. From Grandell[14], we know that the compound Poisson processes can be approximated by the following diffusion process :

$ \begin{align} \mathrm{d}(\sum\limits_{i=1}^{N_{1}(t)+N(t)} X_{i})=a_{1} \mathrm{\; d} t-\sigma_{1} \mathrm{\; d} B_{1}(t), \; \mathrm{d}(\sum\limits_{i=1}^{N_{2}(t)+N(t)} Y_{i})=a_{2} \mathrm{\; d} t-\sigma_{2} \mathrm{\; d} B_{2}(t), \end{align} $ (2.2)

where $ a_{1}=\mu_{1}\left(\lambda_{1}+\lambda\right) , \sigma_{1}^{2}= \nu_{1}\left(\lambda_{1}+\lambda\right) , a_{2}=\mu_{2}\left(\lambda_{2}+\lambda\right) , \sigma_{2}^{2}= \nu_{2}\left(\lambda_{2}+\lambda\right). $ Here $ B_{1}(t) $ and $ B_{2}(t) $ are standard Brownian motions with correlation coefficient $ \rho_{0} \in(-1, 1) . $ To be protected from potential large claims, the insurer is allowed to purchase proportional reinsurance to disperse risk, let $ q_{1}(t), q_{2}(t) \in[0, 1] $ be the reinsurance retention levels for the $ 1 $th and $ 2 $th line of business at time $ t $, respectively. Then by Liang and Yuen[15], the dynamics of $ X^{q}(t) $ can be described by

$ \begin{align} \begin{aligned} \mathrm{d} X^{q}(t)=&\left(c-c_{1}\right)\mathrm{d}t-q_{1}(t) \mathrm{d}(\sum\limits_{i=1}^{N_{1}(t)+N(t)} X_{i})-q_{2}(t) \mathrm{d}(\sum\limits_{i=1}^{N_{2}(t)+N(t)} Y_{i})\\ =&\left[c-c_{1}- a_{1} q_{1}(t)-a_{2} q_{2}(t)\right] \mathrm{d} t+\sqrt{ \sigma_{1}^{2} q_{1}^{2}(t)+\sigma_{2}^{2} q_{2}^{2}(t)+2 q_{1}(t) q_{2}(t) \lambda \mu_{1} \mu_{2} }\mathrm{\; d} W_{0}(t), \end{aligned} \end{align} $ (2.3)

where $ c_{1}=\left(1+\theta\right)\left[\mu_{1}\left(\lambda_{1}+\lambda\right)\left(1-q_{1}(t)\right) +\mu_{2}\left(\lambda_{2}+\lambda\right)\left(1-q_{2}(t)\right)\right] $ is the reinsurance premium rate, $ W_{0}(t) $ is a standard Brownian motion.

2.2 Financial Market

The insurer is assumed to invest in a risk-free asset whose price process is governed by $ \mathrm{d} B(t)=r B(t) \mathrm{d} t, B(0)=b_{0} $ and a risky asset whose price process follows Heston model:

$ \begin{align} \begin{array}{l} \mathrm{d} S(t)=S(t)\left[(r+\alpha L(t)) \mathrm{d} t+\sqrt{L(t)} \mathrm{d} W_{1}(t)\right], S(0)=s_{0}; \end{array} \end{align} $ (2.4)
$ \begin{align} \begin{array}{l} \mathrm{\; d} L(t)=k(m-L(t)) \mathrm{d} t+\sigma \sqrt{L(t)} \mathrm{d} \bar{W}_{2}(t), L(0)=l_{0}, \end{array} \end{align} $ (2.5)

where positive constant $ r $ is the risk-free interest rate, $ \alpha, k, m $, and $ \sigma $ are all positive constants, and $ W_{1}(t) $ and $ \bar{W}_{2}(t) $ are two standard Brownian motions with $ E\left[W_{1}(t) \bar{W}_{2}(t)\right]=\rho t, \rho \in $ $ [-1, 1] . $ By standard Gaussian linear regression, $ \bar{W}_{2}(t) $ can be rewritten as

$ \begin{align} \mathrm{d} \bar{W}_{2}(t)=\rho \mathrm{d} W_{1}(t)+\sqrt{1-\rho^{2}} \mathrm{\; d} W_{2}(t), \end{align} $ (2.6)

where $ W_{2}(t) $ is another standard Brownian motion. We assume that $ W_{0}(t), W_{1}(t) $ and $ W_{2}(t) $ are mutually independent. Moreover, we require $ 2 k m \geq \sigma^{2} $ to ensure that $ L(t) $ is almost surely nonnegative. Similar to Bi et al.[12], we assume that there exists a defaultable bond with a maturity date $ T_{1} $ and that the default time is $ \tau $. The dynamics of the defaultable bond price under $ P $ is given by

$ \begin{align} \mathrm{d} p\left(t, T_{1}\right)=p\left(t-, T_{1}\right)\left[r \mathrm{d} t+(1-Z(t)) \delta(1-\Delta) \mathrm{d} t-(1-Z(t-)) \varsigma \mathrm{d} M^{P}(t)\right], \end{align} $ (2.7)

where $ \delta=h^{Q} \varsigma $ is the risk neutral credit spread, $ 1 / \Delta \geq 1 $ is the default risk premium, $ \varsigma \in[0, 1] $ is the loss rate of the bond when a default occurs, $ M^{P}(t)=Z(t)-\int_{0}^{1}(1-Z(s))h^{p} \mathrm{d}s $ is a martingale and $ Z(t)={\bf 1}_{\{\tau \leq t\}} $ is a default process.

2.3 Wealth Process with Delay

Let $ Y^{\pi}(t), \bar{Y}^{\pi}(t) $ and $ G^{\pi}(t) $ be the delayed wealth and average and pointwise performance of the wealth in the past horizon $ [t-h, t] $ respectively, i.e.

$ \begin{align} Y^{\pi}(t)=\int_{-h}^{0} e^{\beta s} X^{\pi}(t+s) \mathrm{d} s , \; \bar{Y}^{\pi}(t)=\frac{Y^{\pi}(t)}{\int_{-h}^{0} e^{\beta s} \mathrm{\; d} s}, \; G^{\pi}(t)=X^{\pi}(t-h), \end{align} $ (2.8)

for $ \forall t \in[0, T] $, where $ \beta \geq 0 $ is an average parameter and $ h> $ 0 is the delay parameter. Denote by the function $ f\left(t, X^{\pi}(t)-\bar{Y}^{\pi}(t), X^{\pi}(t)-G^{\pi}(t)\right) $ the capital inflow/outflow amount, which is proportional to the past performance of the insurer's wealth, i.e.

$ \begin{align} \begin{aligned} f\left(t, X^{\pi}(t)-\bar{Y}^{\pi}(t), X^{\pi}(t)-G^{\pi}(t)\right)&=k_{1}\left(X^{\pi}(t)-\bar{Y}^{\pi}(t)\right)+k_{2}\left(X^{\pi}(t)-G^{\pi}(t)\right)\\ &=\left(k_{1}+k_{2}\right)X^{\pi}(t)-k_{1}\bar{Y}^{\pi}(t)-k_{2}G^{\pi}(t), \end{aligned} \end{align} $ (2.9)

where $ k_{1} $ and $ k_{2} $ are two nonnegative constants, $ X^{\pi}(t)-G^{\pi}(t) $ represents the absolute performance of wealth, $ X^{\pi}(t)-\bar{Y}^{\pi}(t) $ stands for the average performance of the wealth. Let $ \pi(t):= $ $\{ \left.q_{1}(t), q_{2}(t), \pi_{1}(t), \pi_{2}(t)\right\}_{t \in[0, T]} $ be the reinsurance-investment strategy. Following Shen et al.[9] and A et al.[10], when we consider the capital inflow/outflow function, the stochastic diferential delay equation (SDDE) can be given as follows:

$ \begin{align} \begin{aligned} \mathrm{d} X^{\pi}(t)=&\mathrm{d} X^{q}(t)+\pi_{1}(t) \frac{\mathrm{d} S(t)}{S(t)}+\pi_{2}(t) \frac{\mathrm{d} p\left(t, T_{1}\right)}{p\left(t-, T_{1}\right)}+\left(X^{\pi}(t)-\pi_{1}(t)-\pi_{2}(t)\right) \frac{\mathrm{d} B(t)}{B(t)}\\&-f\left(t, X^{\pi}(t)-\bar{Y}^{\pi}(t), X^{\pi}(t)-G^{\pi}(t)\right) \mathrm{d} t\\ =&\left[A X^{\pi}(t)+c-c_{1}-a_{1}q_{1}(t)-a_{2}q_{2}(t)+\alpha l \pi_{1}(t)+\delta(1-z)(1-\Delta)\pi_{2}(t)\right.\\ &\left.+\bar{k}_{1} Y^{\pi}(t)+k_{2} G^{\pi}(t)\right]\mathrm{d} t+\pi_{1}(t) \sqrt{l} \mathrm{d} W_{1}(t)-\pi_{2}(t)(1-Z(t-)) \varsigma \mathrm{d} M^{P}(t)\\ &+\sqrt{\sigma_{1}^{2}q_{1}^{2}(t)+\sigma_{2}^{2}q_{2}^{2}(t)+2 \lambda \mu_{1}\mu_{2}q_{1}(t)q_{2}(t)} \mathrm{d} W_{0}(t), \end{aligned} \end{align} $ (2.10)

where

$ \begin{align} A=r-k_{1}-k_{2}, \; \; \bar{k}_{1}=\frac{k_{1}}{\int_{-h}^{0} e^{\beta s} \mathrm{\; d} s}. \end{align} $ (2.11)

Furthermore, the initial value of the average performance wealth is $ Y^{\pi}(0)=\frac{x_{0}\left(1-e^{-\beta h}\right)}{\beta} $.

Definition 1 (admissible strategy) For any fixed $ t \in[0, T] $, the strategy $ \pi $ is said to be admissible if it is $ F_{t} $-progressively measurable and satisfies

(ⅰ) $ q_{i}(t) \in\left[0, 1\right], i=1, 2 $;

(ⅱ) $ E\left(\int_{0}^{T}\|\pi(s)\|^{2} d s\right)<\infty, $ where $ \|\pi(t)\|^{2}=q_{1}^{2}(t)+q_{2}^{2}(t)+\pi_{1}^{2}(t)+\pi_{2}^{2}(t) $;

(ⅲ) $ \forall(t, x, y, l) \in[0, T] \times R \times R \times R^{+}, $ the SDDE (2.10) has a pathwise unique solution $ \left\{X^{\pi}(t), t \in[0, T]\right\} $ with $ E_{t, x, y, l}^{Q^{*}} U\left[\left(X^{\pi}(T), Y^{\pi}(T)\right)\right]<\infty, $ where $ Q^{*} $ is the chosen model to describe the worst case, $ E_{t, x, y, l}[\cdot] $ is the condition expectation given $ X^{\pi}(t)=x, Y^{\pi}(t)=y, L(t)=l $.

3 Robust Optimal Control Problem for an AAI

We assume that the insurer is concerned with $ X^{\pi}(T) $ and $ Y^{\pi}(T) $ in the time interval $ [T-h $, $ T] $. Therefore, the insurer has the following exponential utility function:

$ \begin{align} U\left(X^{\pi}(T), Y^{\pi}(T)\right)=-\frac{1}{n} e^{-n\left(X^{\pi}(T)+\varepsilon Y^{\pi}(T)\right)}. \end{align} $ (3.1)

For simplicity, we call the combination $ X^{\pi}(T)+\varepsilon Y^{\pi}(T) $ the terminal wealth. Traditionally, the insurer is assumed to be an ambiguity-neutral investor (ANI) with objective function as

$ \begin{align} \begin{array}{l} \mathop{sup}\limits_{\pi \in \Pi} E\left[U\left(X^{\pi}(T), Y^{\pi}(T)\right) \mid X^{\pi}(t)=x, Y^{\pi}(t)=\right.y, L(t)=l], \end{array} \end{align} $ (3.2)

where $ \Pi $ is the set of all admissible strategies. Actually, the insurer would like to consider the alternative probability measure rather than reference probability measure P because of uncertainty on the parameters. The alternative models are defined by $ \mathrm{Q}:=\{Q \mid Q \sim P\} $. Next, define a process $ \left\{\theta(t)=\left(\theta_{0}(t), \theta_{1}(t), \theta_{2}(t)\right) \mid t \in\right. $ $ [0, T]\} $ satisfying that (ⅰ) $ \theta(t) $ is $ F_{t} $ -measurable, $ \forall t \in[0, T] $; (ⅱ) $ E\left[\exp \left\{(1 / 2) \int_{0}^{T}\|\theta(t)\|^{2} \mathrm{\; d} t\right\}\right]<\infty, $ where $ \|\theta(t)\|^{2}= $ $ \sum_{i=0}^{2} \theta_{i}^{2}(t) $. We denote $ \Theta $ for the space of all such processes $ \theta $. For $ \forall \theta \in \Theta $, we define a real-valued process $ \left\{\Lambda^{\theta}(\mathrm{t}) \mid t \in\right. $ $ [0, T]\} $ on $ (\Omega, F, P) $ by

$ \begin{align} \Lambda^{\theta}(\mathrm{t})=\left\{-\int_{0}^{t} \theta(u) \mathrm{d} W(u)-\frac{1}{2} \int_{0}^{t}\|\theta(u)\|^{2} \mathrm{\; d} u\right\}, \end{align} $ (3.3)

where $ W(t)=\left(W_{0}(t), W_{1}(t), W_{2}(t)\right)^{\prime} $. By it's differentiation rule,

$ \begin{align} \mathrm{d} \Lambda^{\theta}(t)=\Lambda^{\theta}(t)[-\theta(t) \mathrm{d} W(t)]. \end{align} $ (3.4)

Thus $ \Lambda^{\theta}(t) $ is a $ P $-martingale. Hence, $ E\left[\Lambda^{\theta}(t)\right]=1 $. For $ \forall \theta \in \Theta $, a new real-word probability measure $ Q $ absolutely continuous to $ P $ on $ F_{T} $ is defined by$ \left.\frac{\mathrm{d} Q}{\mathrm{\; d} P}\right|_{\mathrm{F}_{T}}:=\Lambda^{\theta}(t) $. So far, we have constructed a family of real-world probability measures $ Q $ parameterized by $ \theta \in \Theta $. Applying Girsanov's theorem, we can see that

$ \begin{align} \begin{array}{l} \mathrm{d} W_{i}^{\mathrm{Q}}(t)=\mathrm{d} W_{i}(t)+\theta_{i}(t)\mathrm{d} t, \; i=0, 1, 2. \end{array} \end{align} $ (3.5)

Thus, the stochastic process (2.5) under Q is

$ \begin{align} \begin{aligned} \mathrm{d} L(t)=&\left[r(m-L(t))-\sigma \rho \theta_{1}(t) \sqrt{L(t)}-\sigma \theta_{2}(t) \sqrt{L(t)} \sqrt{1-\rho^{2}}\right] \mathrm{d} t\\ &+\sigma \rho \sqrt{L(t)} \mathrm{d} W_{1}^{\mathrm{Q}}(t)+\sigma \sqrt{L(t)} \sqrt{1-\rho^{2}} \mathrm{\; d} W_{2}^{\mathrm{Q}}(t). \end{aligned} \end{align} $ (3.6)

And the wealth process (2.10) under $ Q $ is rewritten as

$ \begin{align} \begin{aligned} \mathrm{d} X^{\pi}(t)=&\left[A X^{\pi}(t)+(\eta-\theta)(a_{1}+a_{2})+\theta[a_{1}q_{1}(t)+a_{2}q_{2}(t)]+\alpha l \pi_{1}(t)+\delta(1-z)(1-\Delta)\pi_{2}(t)\right.\\&\left.+\bar{k}_{1} Y^{\pi}(t)+k_{2} G^{\pi}(t)-\theta_{0}(t)\sqrt{\sigma_{1}^{2}q_{1}^{2}(t)+\sigma_{2}^{2}q_{2}^{2}(t)+2 \lambda \mu_{1}\mu_{2}q_{1}(t)q_{2}(t)}\right.\\&\left.-\theta_{1}(t)\pi_{1}(t) \sqrt{l} \right]\mathrm{d} t+\sqrt{\sigma_{1}^{2}q_{1}^{2}(t)+\sigma_{2}^{2}q_{2}^{2}(t)+2 \lambda \mu_{1}\mu_{2}q_{1}(t)q_{2}(t)} \mathrm{d} W_{0}^{Q}(t)\\ &+\pi_{1}(t) \sqrt{l} \mathrm{d} W_{1}^{Q}(t)-\pi_{2}(t)(1-Z(t-)) \varsigma \mathrm{d} M^{P}(t). \end{aligned} \end{align} $ (3.7)

Inspired by Zhang and Chen[6], we formulate the following robust control problem to modify problem (3.2), i.e.

$ \begin{align} \begin{aligned} V(t, x, y, l, z)=\sup _{\pi \in \Pi}\inf _{\theta \in \Theta} E^{Q}\left[U\left(X^{\pi}(T), Y^{\pi}(T)\right)+\int_{t}^{T} \Psi\left(s, X^{\pi}(s), Y^{\pi}(s), \theta(s)\right) \mathrm{d} s\right], \end{aligned} \end{align} $ (3.8)

where

$ \begin{align} \Psi(t)=\frac{\theta_{0}^{2}(t)}{2 \psi_{0}(t)}+\frac{\theta_{1}^{2}(t)}{2 \psi_{1}(t)}+\frac{\theta_{2}^{2}(t)}{2 \psi_{2}(t)}+ h^{P}(1-Z(t))\frac{\theta_{3}(t)\ln \theta_{3}(t)-\theta_{3}(t)+1}{\psi_{3}(t)}, \end{align} $ (3.9)

$ \psi_{i}(t)=-\frac{\beta_{i}}{nV(t, x, y, l, z)}, i=0, 1, \ldots, 3, $ and $ \beta_{i}>0, i=0, 1, \ldots, 3 $ are the ambiguity-aversion coefficients representing the ambiguity-averse level of the insurer to the model uncertainty. In order to solve the optimization problem (3.8) and derive the optimal strategy $ \pi^*(t) $. By dynamic programming principle, the HJB equation for (3.8) can be derived as

$ \begin{align} \sup _{\pi \in \Pi}\inf _{\theta \in \Theta}\left\{L^{\pi, \theta} V+\frac{\theta_{0}^{2}}{2 \psi_{0}}+\frac{\theta_{1}^{2}}{2 \psi_{1}}+\frac{\theta_{2}^{2}}{2 \psi_{2}}+ h^{P}(1-z)\frac{\theta_{3}\ln \theta_{3}-\theta_{3}+1}{\psi_{3}}\right\}=0, \end{align} $ (3.10)

with the boundary condition $ V(t, x, y, l, z)=U\left(X^{\pi}(T), Y^{\pi}(T)\right), $where $ L^{\pi, \theta} $ is the variational operator and is defined by

$ \begin{align} \begin{aligned} L^{\pi, \theta} V =&V_{t}+ V_{x}\left[Ax+(\eta-\theta)(a_{1}+a_{2})+\theta[a_{1}q_{1}+a_{2}q_{2}]+\alpha l \pi_{1}+\delta(1-z)(1-\Delta)\pi_{2}+\bar{k}_{1}y\right.\\&\left.+k_{2} g-\theta_{0}\sqrt{\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}}-\theta_{1}\pi_{1} \sqrt{l} \right]+ V_{y}\left(x-\beta y-e^{-\beta h}g\right)\\& + V_{l}\left[k(m-l)-\sigma \rho \theta_{1} \sqrt{l}-\sigma \theta_{2} \sqrt{l} \sqrt{1-\rho^{2}}\right]+\frac{1}{2} V_{x x}\left[\pi_{1}^{2} l+\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}\right] \\ &+\frac{1}{2} V_{l l}\sigma^{2} l +V_{x l}\sigma\rho l \pi_{1} + \theta_{3} h^{P}(1-z)\left[V\left(t, x-\pi_{2} \varsigma, y, l, 1\right)-V(t, x, y, l, 0)\right], \end{aligned} \end{align} $ (3.11)

here $ V_{t}, V_{x}, V_{y}, V_{l}, V_{x x}, V_{l l} $ and $ V_{x l} $ represent the partial derivatives of value function with respect to the corresponding variables.

4 Robust Optimal Investment and Reinsurance Strategies

In the following, we set out to derive the explicit solutions to the HJB equation (3.10) for the post-default case and the pre-default case, respectively.

4.1 Post-Default Case

In the post-default case, i.e. $ z = 1 $, we have that $ p\left(t, T_{1}\right)=0, \tau \leq t \leq T $. Thus $ \pi_{2}(t)=0 $ for $ \tau \leq t \leq T $. Then the HJB equation (3.10) can be rewritten as the following form:

$ \begin{align} \begin{aligned} \sup _{\pi \in \Pi}\inf _{\theta \in \Theta}&\left\{ V_{t}+ V_{x}\left[Ax+(\eta-\theta)(a_{1}+a_{2})+\theta[a_{1}q_{1}+a_{2}q_{2}]+\alpha l \pi_{1}+\bar{k}_{1}y+k_{2} g-\theta_{1}\pi_{1} \sqrt{l}\right.\right.\\ &\left.-\theta_{0}\sqrt{\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}} \right]+ V_{y}\left(x-\beta y-e^{-\beta h}g\right)+ V_{l}\left[k(m-l)-\sigma \rho \theta_{1} \sqrt{l}\right.\\ &\left.-\sigma \theta_{2} \sqrt{l} \sqrt{1-\rho^{2}}\right]+\frac{1}{2} V_{x x}\left[\pi_{1}^{2} l+\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}\right]+\frac{1}{2} V_{l l}\sigma^{2} l +V_{x l}\sigma\rho l \pi_{1}\\ &\left. -\frac{n V}{2 \beta_{0}}\theta_{0}^{2}-\frac{n V}{2 \beta_{1}}\theta_{1}^{2}-\frac{n V}{2 \beta_{2}}\theta_{2}^{2}\right\}=0, \end{aligned} \end{align} $ (4.1)

with the boundary condition $ V(T, x, y, l, 1)=-\frac{1}{n} e^{-n\left(x+{\varepsilon}y\right)}. $ According to A and Li[10], we assume the parameters satisfy the following conditions:

$ \begin{align} \begin{array}{l} k_{2}=\varepsilon e^{-\beta h} , \; \bar{k}_{1}=(\beta+A+\varepsilon) \varepsilon. \end{array} \end{align} $ (4.2)

According to (4.1) and the boundary condition of $ V(T, x, y, l, 1) $, we conjecture that the optimal value function has the following form:

$ \begin{align} V(T, x, y, l, 1)=-\frac{1}{n} e^{-n\left(x+{\varepsilon}y\right)H(t)+M(t)l+N(t)}. \end{align} $ (4.3)

The boundary condition implies that $ H(T)=1, M(T)=0, N(T)=0. $ We get partial derivative

$ \begin{align} \begin{array}{l} V_{t}=\left[-n\left(x+{\varepsilon}y\right) H^{'}(t)+ M^{'}(t)l+ N^{'}(t)\right]V, \; V_{x}=-nH(t)V, \; V_{y}=-n\varepsilon H(t)V \\ V_{l}=M(t)V, \; V_{x x}=n^{2}H^{2}(t)V , \; V_{l l}=M^{2}(t)V , \; V_{x l}=-nH(t)M(t)V. \end{array} \end{align} $ (4.4)

Inserting (4.4) into (4.1), fix $ \pi $ and maximize over $ \theta $, by the first-order conditions we get

$ \begin{align} \begin{array}{l} \theta_{0}^{*}(t)=\beta_{0}H(t)\sqrt{\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}}\; , \\ \theta_{1}^{*}(t)=\beta_{1}H(t)\pi_{1}\sqrt{l}-\frac{\beta_{1}}{n}M(t)\sigma\rho\sqrt{l}\; , \\ \theta_{2}^{*}(t)=-\frac{\beta_{2}}{n}M(t)\sigma\sqrt{l}\sqrt{1-\rho^{2}}\; . \end{array} \end{align} $ (4.5)

Inserting (4.4) and (4.5) into (4.1), by the first-order conditions we get

$ \begin{align} \begin{array}{l} \pi_{1}^{*}(t)=\frac{1}{H(t)}\left[\frac{\alpha}{n+\beta_{1}}+\frac{\sigma \rho}{n}M(t)\right], \quad \pi_{2}^{*}(t)=0\; . \end{array} \end{align} $ (4.6)

Taking the above conditions into (4.1), we get

$ \begin{align} \begin{aligned} &-n\left(x+{\varepsilon}y\right) H^{'}(t)+ M^{'}(t)l+ N^{'}(t)-nH(t)\left[Ax+(\eta-\theta)(a_{1}+a_{2})+\bar{k}_{1}y+k_{2} g\right]-n\varepsilon H(t)\\ &\left(x-\beta y-e^{-\beta h}g\right)+r(m-l)M(t)+\frac{1}{2}\left[1 +\frac{\beta_{2}}{n}\right]l(1-\rho^{2}) \sigma^{2}M^{2}(t)-\frac{nl\alpha^{2}}{2(n+\beta_{1})}-\alpha l \sigma\rho M(t)\\ &+\inf\limits_{q_{1}, q_{2}}\left\{f(q_{1}, q_{2}, t)\right\}=0, \end{aligned} \end{align} $ (4.7)

where $ f(q_{1}, q_{2}, t)=-n\theta(a_{1}q_{1}+a_{2}q_{2})H(t)+\frac{n(n+\beta_{0})}{2} (\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2})H^{2}(t) $. For any t ∈ [0, T], we get the derivative of $ f(q_{1}, q_{2}, t) $ with respect to (w.r.t.) $ q_{1} $ and $ q_{2} $

$ \begin{align} \begin{array}{l} \frac{\partial f}{\partial q_1}=-n\theta a_1 H(t)+n(n+\beta_0)[\sigma_1^2 q_1+\lambda\mu_1\mu_2q_2] H^2(t), \frac{\partial^2 f}{\partial q_1^2}=n(n+\beta_0)\sigma_1^2 H^2(t), \\ \frac{\partial f}{\partial q_2}=-n\theta a_2 H(t)+n(n+\beta_0)[\sigma_2^2 q_2+\lambda\mu_1\mu_2q_1] H^2(t), \frac{\partial^2 f}{\partial q_2^2}=n(n+\beta_0)\sigma_2^2 H^2(t), \\ \frac{\partial^2 f}{\partial q_1 \partial q_2}=\frac{\partial^2 f}{\partial q_2 \partial q_1}=n(n+\beta_0)\lambda\mu_1\mu_2 H^2(t). \end{array} \end{align} $ (4.8)

So we have following Hessian matrix

$ \begin{align} \left[ \begin{array}{cc} \frac{\partial^2 f}{\partial q_1^2}&\frac{\partial^2 f}{\partial q_1 \partial q_2}\\ \frac{\partial^2 f}{\partial q_2 \partial q_1}&\frac{\partial^2 f}{\partial q_2^2}\\ \end{array} \right]=n(n+\beta_0) H^2(t)B, \; B=\left[ \begin{array}{cc} \sigma_1^2&\lambda\mu_1\mu_2\\ \lambda\mu_1\mu_2&\sigma_2^2\\ \end{array} \right]. \end{align} $ (4.9)

By the Lemma 1 of Yuen et al.[13], B is a positive definite matrix, which means that the Hessian matrix is also a positive definite matrix. Now substituting $ q^*(t)=\left(q_1^*(t), q_1^*(t)\right) $ into (4.7) and separating the variables, we derive the following equations:

$ \begin{align} \begin{array}{l} \left(x+{\varepsilon}y\right) H^{'}(t)+\left[Ax+\bar{k}_{1}y+k_{2} g+\varepsilon \left(x-\beta y-e^{-\beta h}g\right)\right] H(t)=0, \end{array} \end{align} $ (4.10)
$ \begin{align} \begin{array}{l} M^{'}(t)-(k+\alpha\sigma\rho)M(t)+\frac{n+\beta_2}{2n}\sigma^2(1-\rho^2)M^2(t)-\frac{n\alpha^2}{2(n+\beta_1)}=0, \end{array} \end{align} $ (4.11)
$ \begin{align} \begin{array}{l} N^{'}(t)+k m M(t)-n(\eta-\theta)(a_1+a_2)H(t)+f(q_{1}^*, q_{2}^*, t)=0, \end{array} \end{align} $ (4.12)

with the boundary conditions $ H(T)=1, M(T)=0, N(T)=0 $. Solving (4.10) and (4.11), we get

$ \begin{align} \begin{array}{l} H(t)=e^{(A+\varepsilon)(T-t)}, 0\leq t \leq T, \end{array} \end{align} $ (4.13)
$ \begin{align} \begin{array}{l} M(t)=\frac{b_{3}\left[1-e^{b_{1}(T-t)}\right]}{2 b_{1}+\left(b_{1}+b_{2}\right)\left[e^{b_{1}(T-t)}-1\right]}, \end{array} \end{align} $ (4.14)

where $ b_{1}=\sqrt{\left(k+\alpha\sigma\rho\right)^{2}+\frac{n+\beta_{2}}{n+\beta_{1}} \sigma^{2} \alpha^{2}(1-\rho^{2})} $, $ b_{2}=k+\alpha\sigma\rho, b_{3}=\frac{n \alpha^{2}}{n+\beta_{1}} $.

The Hessian matrix (4.9) is a positive definite matrix, thus $ f(q_{1}, q_{2}, t) $ is a convex function w.r.t. $ q_{1} $ and $ q_{2} $. Therefore, using the first-order optimization conditions, we obtain

$ \begin{align} \begin{array}{l} \hat{q}_1(t)=\frac{\theta D_1}{(n+\beta_0)e^{(A+\varepsilon)(T-t)} D_3}, \hat{q}_2(t)=\frac{\theta D_2}{(n+\beta_0)e^{(A+\varepsilon)(T-t)} D_3}, \end{array} \end{align} $ (4.15)

where $ D_{1}=a_{1} \sigma_{2}^{2}-a_{2} \lambda \mu_{1} \mu_{2} , D_{2}=a_{2} \sigma_{1}^{2}-a_{1} \lambda \mu_{1} \mu_{2} , D_{3}=\sigma_{1}^{2} \sigma_{2}^{2}-\lambda^{2} \mu_{1}^{2} \mu_{2}^{2}. $ Note that $ D_{1}, D_{2}, D_{3}>0, $ thus $ \hat{q}_{1}(t)>0, \hat{q}_{2}(t)>0 $.

Furthermore, let

$ \begin{align} t_{1}=\left\{\begin{array}{lc}T, & \theta D_{1} \leq\left(n+\beta_{0}\right) D_{3}, \\ T-\frac{1}{A+\varepsilon} \ln \frac{\theta D_{1}}{\left(n+\beta_{0}\right) D_{3}}, &\left(n+\beta_{0}\right) D_{3}<\theta D_{1}<\left(n+\beta_{0}\right) D_{3} e^{(A+\varepsilon)T}, \\ 0, & \theta D_{1} \geq\left(n+\beta_{0}\right) D_{3} e^{(A+\varepsilon)T}, \end{array}\right. \end{align} $ (4.16)
$ \begin{align} t_{2}=\left\{\begin{array}{lc}T, & \theta D_{2} \leq\left(n+\beta_{0}\right) D_{3}, \\ T-\frac{1}{A+\varepsilon} \ln \frac{\theta D_{2}}{\left(n+\beta_{0}\right) D_{3}}, &\left(n+\beta_{0}\right) D_{3}<\theta D_{2}<\left(n+\beta_{0}\right) D_{3} e^{(A+\varepsilon)T}, \\ 0, & \theta D_{2} \geq\left(n+\beta_{0}\right) D_{3} e^{(A+\varepsilon)T}, \end{array}\right. \end{align} $ (4.17)

We now discuss the optimal reinsurance strategy and N(t) in different cases.

Case 1  $ D_{1} \leq D_{2}, \; then\; t_{1} \geq t_{2}. $

(1) When $ 0 \leq t<t_{2}, $ the optimal reinsurance proportion is $ q^{*}(t)=\left(\hat{q}_{1}(t), \hat{q}_{2}(t)\right), $ where $ \hat{q}_{1}(t) $ and $ \hat{q}_{2}(t) $ are given by (4.15). So we can obtain

$ \begin{align} \begin{array}{l} N_1(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(\hat{q}_{1}(s), \hat{q}_{2}(s)\right)\right] \mathrm{d} s+e_{1}, \end{array} \end{align} $ (4.18)

(2) When $ t \geq t_{2}, $ we have $ \hat{q}_{2}(t) \geq 1, $ so $ q_{2}^{*}(t)=1 . $ Substituting $ q_{2}^{*}(t)=1 $ into $ \inf\limits_{q_{1}, q_{2}}\left\{f(q_{1}, q_{2}, t)\right\} $ and by the first-order conditions we get

$ \begin{align} \begin{array}{l} \bar{q}_{1}(t)=\frac{\theta a_{1}-\lambda \mu_{1} \mu_{2}\left(n+\beta_{0}\right) e^{(A+\varepsilon)(T-t)}}{\sigma_{1}^{2}+\left(n+\beta_{0}\right) e^{(A+\varepsilon)(T-t)}}. \end{array} \end{align} $ (4.19)

Let

$ \begin{align} \begin{array}{l} t_{1}^{\prime}=T-\frac{1}{(A+\varepsilon)} \ln \frac{\theta a_{1}}{\left(n+\beta_{0}\right)\left(\sigma_{1}^{2}+\lambda \mu_{1} \mu_{2}\right)}, \end{array} \end{align} $ (4.20)

then for $ t_{2} \leq t < t_{1}^{\prime}, $ the optimal reinsurance strategy is $ q^{*}(t)=\left(\bar{q}_{1}(t), 1\right). $ So,

$ \begin{align} \begin{array}{l} N_2(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(\bar{q}_{1}(s), 1\right)\right] \mathrm{d} s+e_{2}, \end{array} \end{align} $ (4.21)

(3) For $ t_{1}^{\prime} \leq t \leq T, $ it is easy to see that $ q^{*}(t)=(1, 1) $. so we can obtain

$ \begin{align} \begin{array}{l} N_3(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(1, 1\right)\right] \mathrm{d} s. \end{array} \end{align} $ (4.22)

Case 2  $ D_{1} > D_{2} , \; then\; t_{1}<t_{2}. $

(1) When $ 0 \leq t<t_{1}, $ the optimal reinsurance proportion is $ q^{*}(t)=\left(\hat{q}_{1}(t), \hat{q}_{2}(t)\right), $ where $ \hat{q}_{1}(t) $ and $ \hat{q}_{2}(t) $ are given by (4.15). So we can obtain

$ \begin{align} \begin{array}{l} N_4(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(\hat{q}_{1}(s), \hat{q}_{2}(s)\right)\right] \mathrm{d} s+e_{3}, \end{array} \end{align} $ (4.23)

(2) When $ t \geq t_{1}, $ we have $ \hat{q}_{1}(t) \geq 1, $ so $ q_{1}^{*}(t)=1 . $ Substituting $ q_{1}^{*}(t)=1 $ into $ \inf\limits_{q_{1}, q_{2}}\left\{f(q_{1}, q_{2}, t)\right\} $ and by the first-order conditions we get

$ \begin{align} \begin{array}{l} \bar{q}_{2}(t)=\frac{\theta a_{2}-\lambda \mu_{1} \mu_{2}\left(n+\beta_{0}\right) e^{(A+\varepsilon)(T-t)}}{\sigma_{1}^{2}+\left(n+\beta_{0}\right) e^{(A+\varepsilon)(T-t)}}. \end{array} \end{align} $ (4.24)

Let

$ \begin{align} \begin{array}{l} t_{2}^{\prime}=T-\frac{1}{(A+\varepsilon)} \ln \frac{\theta a_{2}}{\left(n+\beta_{0}\right)\left(\sigma_{2}^{2}+\lambda \mu_{1} \mu_{2}\right)}, \end{array} \end{align} $ (4.25)

then for $ t_{1} \leq t < t_{2}^{\prime}, $ the optimal reinsurance strategy is $ q^{*}(t)=\left(1, \bar{q}_{2}(t)\right). $ So,

$ \begin{align} \begin{array}{l} N_5(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(1, \bar{q}_{2}(s)\right)\right] \mathrm{d} s+e_{4}, \end{array} \end{align} $ (4.26)

(3) For $ t_{2}^{\prime} \leq t \leq T, $ it is easy to see that $ q^{*}(t)=(1, 1) $, so we can obtain

$ \begin{align} \begin{array}{l} N_3(t)=\int_{t}^{T}\left[-n (\eta-\theta)(a_1+a_2)H(s)+kmM(s)+f\left(1, 1\right)\right] \mathrm{d} s. \end{array} \end{align} $ (4.27)

where $ e_{1}=\int_{t_{2}}^{T}\left[f\left(\bar{q}_{1}(s), 1\right)-f\left(\hat{q}_{1}(s), \hat{q}_{2}(s)\right)\right] \mathrm{d} s+e_{2} $, $ e_{2}=\int_{t_{1}^{\prime}}^{T}\left[f(1, 1)-f\left(\bar{q}_{1}(s), 1\right)\right] \mathrm{d} s $, $ e_{3}=\int_{t_{1}}^{T}\left[f\left(1, \bar{q}_{2}(s)\right)-f\left(\hat{q}_{1}(s), \hat{q}_{2}(s)\right)\right] \mathrm{d} s+e_{4} $ and $ e_{4}=\int_{t_{2}^{\prime}}^{T}\left[f(1, 1)-f\left(1, \bar{q}_{2}(s)\right)\right] \mathrm{d}s. $

Summarizing the above results, we can obtain the following theorem.

Theorem 4.1  In the post-default case, for problem (3.8) with the exponential utility function (3.1), the robust optimal investment strategies are given by

$ \begin{array}{l} \pi_{1}^{*}(t)=\frac{1}{H(t)}\left[\frac{\alpha}{n+\beta_{1}}+\frac{\sigma \rho}{n}M(t)\right], \\ \pi_{2}^{*}(t)=0. \end{array} $

The robust optimal reinsurance strategy and the optimal value function are given as follows:

(1) When $ D_{1} \leq D_{2} $, the robust optimal reinsurance strategies are

$ \left(q_{1}^{*}(t), q_{2}^{*}(t)\right)=\left\{\begin{array}{l} \left(\hat{q}_{1}(t), \hat{q}_{2}(t)\right), \; 0 \leq t<t_{2}, \\ \left(\bar{q}_{1}(t), 1\right), \; t_{2} \leq t<t_{1}^{\prime}, \\ (1, 1), \; t_{1}^{\prime} \leq t<T, \end{array}\right. $

with $ t_{2} $ and $ t_{1}^{\prime} $ given in equations (4.17) and (4.20). The value function is given by

$ V(t, x, y, l, 1)=\left\{\begin{aligned} -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_1(t)}, \; & 0 \leq t<t_{2}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_2(t)}, \; & t_{2} \leq t<t_{1}^{\prime}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_3(t)}, \; & t_{1}^{\prime} \leq t<T, \end{aligned}\right. $

(2) When $ D_{1}>D_{2} $, the robust optimal reinsurance strategies are

$ \left(q_{1}^{*}(t), q_{2}^{*}(t)\right)=\left\{\begin{array}{l}\left(\hat{q}_{1}(t), \hat{q}_{2}(t)\right), \; 0 \leq t<t_{1}, \\ \left(1, \bar{q}_{2}(t)\right), \; t_{1} \leq t<t_{2}^{\prime}, \\ (1, 1), \; t_{2}^{\prime} \leq t<T, \end{array}\right. $

with $ t_{1} $ and $ t_{2}^{\prime} $ given in equations (4.16) and (4.25). The value function is given by

$ V(t, x, y, l, 1)=\left\{\begin{aligned} -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_4(t)}, \; & 0 \leq t<t_{1} , \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_5(t)}, \; & t_{1} \leq t<t_{2}^{\prime}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) H(t)+lM(t)+N_3(t)}, \; & t_{2}^{\prime} \leq t<T, \end{aligned}\right. $

with $ H(T), M(t) $ and $ N_{i}(t), i=1, 2, \ldots, 5 $ given in equations $ (4.13), (4.14), (4.18), (4.21)-(4.23) $ and $ (4.26) $, respectively.

4.2 Pre-Default Case

In the pre-default case, i.e. z = 0. Then we can rewrite the HJB equation (3.10) as

$ \begin{align} \begin{aligned} \sup _{\pi \in \Pi}\inf _{\theta \in \Theta}&\left\{ V_{t}+ V_{x}\left[Ax+(\eta-\theta)(a_{1}+a_{2})+\theta[a_{1}q_{1}+a_{2}q_{2}]+\alpha l \pi_{1}+\bar{k}_{1}y+k_{2} g-\theta_{1}\pi_{1} \sqrt{l}\right.\right.\\ &\left.-\theta_{0}\sqrt{\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}} \right]+ V_{y}\left(x-\beta y-e^{-\beta h}g\right)+ V_{l}\left[k(m-l)-\sigma \rho \theta_{1} \sqrt{l}\right.\\ &\left.-\sigma \theta_{2} \sqrt{l} \sqrt{1-\rho^{2}}\right]+\frac{1}{2} V_{x x}\left[\pi_{1}^{2} l+\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}\right]+\frac{1}{2} V_{l l}\sigma^{2} l +V_{x l}\sigma\rho l \pi_{1}\\ &+ \theta_{3} h^{P}\left[V\left(t, x-\pi_{2} \varsigma, y, l, 1\right)-V(t, x, y, l, 0)\right] -\frac{n V}{2 \beta_{0}}\theta_{0}^{2}-\frac{n V}{2 \beta_{1}}\theta_{1}^{2}-\frac{n V}{2 \beta_{2}}\theta_{2}^{2}\\ &\left.-\frac{n V}{ \beta_{3}} h^{P}(\theta_{3}\ln \theta_{3}-\theta_{3}+1)\right\}=0. \end{aligned} \end{align} $ (4.28)

We conjecture the value function

$ \begin{align} V(T, x, y, l, 0)=-\frac{1}{n} e^{-n\left(x+{\varepsilon}y\right)\bar H(t)+\bar M(t)l+\bar N(t)}. \end{align} $ (4.29)

The boundary condition $ V(T, x, y, l, 0)=-\frac{1}{n} e^{-n\left(x+{\varepsilon}y\right)} $ implies that $ \bar H(T)=1, \bar M(T)=0, \bar N(T)=0 $. Similarly to the former subsection, we obtain

$ \begin{align} \begin{array}{l} \theta_{0}^{*}(t)=\beta_{0}\bar H(t)\sqrt{\sigma_{1}^{2}q_{1}^{2}+\sigma_{2}^{2}q_{2}^{2}+2 \lambda \mu_{1}\mu_{2}q_{1}q_{2}}\; , \\ \theta_{1}^{*}(t)=\beta_{1}\bar H(t)\pi_{1}\sqrt{l}-\frac{\beta_{1}}{n}\bar M(t)\sigma\rho\sqrt{l}\; , \\ \theta_{2}^{*}(t)=-\frac{\beta_{2}}{n}\bar M(t)\sigma\sqrt{l}\sqrt{1-\rho^{2}}\; , \\ \theta_{3}^{*}(t)=e^{\frac{\beta_{3}}{n}\left[e^{n\varsigma H(t)\pi_2-n\left(x+{\varepsilon}y\right)(H(t)-\bar H(t))+(M(t)-\bar M(t))l+(N(t)-\bar N(t))}-1\right]}, \end{array} \end{align} $ (4.30)

and the optimal investment strategies

$ \begin{align} \begin{array}{l} \pi_{1}^{*}(t)=\frac{1}{\bar H(t)}\left[\frac{\alpha}{n+\beta_{1}}+\frac{\sigma \rho}{n}\bar M(t)\right], \\ \pi_{2}^{*}(t)=\frac{1}{n \varsigma H(t)}\left[\ln \frac{\delta\bar H(t)}{\theta_3^* h^P \varsigma H(t)}+n\left(x+{\varepsilon}y\right)(H(t)-\bar H(t))-(M(t)-\bar M(t))l-(N(t)-\bar N(t))\right]. \end{array} \end{align} $ (4.31)

Combining $ \theta_3^*(t) $ with $ \pi_2^*(t) $, we further have the following nonlinear equation for $ \theta_3^*(t) $:

$ \begin{align} \begin{array}{l} \frac{n h^P}{\beta_3}\theta_3^*(t)\ln\theta_3^*(t)+h^P \theta_3^*(t)-\frac{\delta\bar H(t)}{\varsigma H(t)}=0. \end{array} \end{align} $ (4.32)

Lemma 1  Nonlinear equation $ \frac{n h^P}{\beta_3}\theta_3^*(t)\ln\theta_3^*(t)+h^P \theta_3^*(t)-\frac{\delta\bar H(t)}{\varsigma H(t)}=0 $ exists a unique positive root $ \theta_3^*(t) $.

Proof  The proof of this lemma is similar to lemma 1 in zhang and chen[6], so we omit it here.

Because the robust optimal reinsurance strategies are same as the post-default case, substituting $ q^*(t), \pi_1^*(t) $ and $ \pi_2^*(t) $ into (4.28) and separating the variables, we obtain

$ \begin{align} \begin{array}{l} \left(x+{\varepsilon}y\right)\bar H^{'}(t)+\left[Ax+\bar{k}_{1}y+k_{2} g+\varepsilon \left(x-\beta y-e^{-\beta h}g\right)\right] \bar H(t)=0, \end{array} \end{align} $ (4.33)
$ \begin{align} \begin{array}{l} \bar M^{'}(t)-(k+\alpha\sigma\rho)\bar M(t)+\frac{n+\beta_2}{2n}\sigma^2(1-\rho^2)\bar M^2(t)-\frac{n\alpha^2}{2(n+\beta_1)}-\frac{\delta \bar H(t)}{\varsigma H(t)}(\bar M(t)-M(t)=0, \end{array} \end{align} $ (4.34)
$ \begin{align} \begin{array}{l} \bar N^{'}(t)+k m \bar M(t)-n(\eta-\theta)(a_1+a_2)\bar H(t)+\frac{\delta \bar H(t)}{\varsigma H(t)}-\theta_3^* h^P-\frac{n h^P}{\beta_3}(\theta_3^*\ln\theta_3^*-\theta_3^*+1)\\ -\frac{\delta \bar H(t)}{\varsigma H(t)}[\ln{\frac{\delta \bar H(t)}{\theta_3^* h^P \varsigma H(t)}}+n\left(x+{\varepsilon}y\right)(H(t)-\bar H(t))]-\frac{\delta \bar H(t)}{\varsigma H(t)}(\bar N(t)-N(t))+f(q_{1}^*, q_{2}^*, t)=0, \end{array} \end{align} $ (4.35)

with the boundary conditions $ \bar H(T)=1, \bar M(T)=0, \bar N(T)=0 $. Similar to (4.13), we can get

$ \begin{align} \begin{array}{l} \bar H(t)=H(t)=e^{(A+\varepsilon)(T-t)}, \; 0\leq t \leq T. \end{array} \end{align} $ (4.36)

Because the solution of nonlinear first order riccati differential equation is unique, we obtain

$ \begin{align} \begin{array}{l} \bar M(t)=M(t)=\frac{b_{3}\left[1-e^{b_{1}(T-t)}\right]}{2 b_{1}+\left(b_{1}+b_{2}\right)\left[e^{b_{1}(T-t)}-1\right]}. \end{array} \end{align} $ (4.37)

Next, let $ G_{i}(t)=\bar N_{i}(t)-N_{i}(t), $ then we have $ G^{'}_{i}(t)=\bar N^{'}_{i}(t)-N^{'}_{i}(t) =\frac{\delta}{\varsigma}G_{i}(t)-\frac{\delta}{\varsigma}+\theta_3^* h^P+\frac{n h^P}{\beta_3}(\theta_3^*\ln\theta_3^*-\theta_3^*+1)+\frac{\delta}{\varsigma}\ln{\frac{\delta}{\theta_3^* h^P \varsigma}}, $ with the boundary conditions $ G_{i}(T)=\bar N_{i}(T)- N_{i}(T)=0 $, we obtain

$ \begin{align} \begin{array}{l} G_{i}(t)=P e^{-\frac{\delta}{\varsigma}(T-t)}-P, \end{array} \end{align} $ (4.38)
$ \begin{align} \begin{array}{l} \bar N_{i}(t)=N_{i}(t)+G_{i}(t)=N_{i}(t)+P e^{-\frac{\delta}{\varsigma}(T-t)}-P, \end{array} \end{align} $ (4.39)

where $ P=-\frac{\delta}{\varsigma}+\theta_3^* h^P+\frac{n h^P}{\beta_3}(\theta_3^*\ln\theta_3^*-\theta_3^*+1)+\frac{\delta}{\varsigma}\ln{\frac{\delta}{\theta_3^* h^P \varsigma}} $.

In the same way, summarizing the above results, we obtain the following theorem.

Theorem 4.2  In the pre-default case, the robust optimal investment strategies are given by

$ \begin{array}{l} \pi_{1}^{*}(t)=\frac{1}{\bar H(t)}\left[\frac{\alpha}{n+\beta_{1}}+\frac{\sigma \rho}{n}\bar M(t)\right], \\ \pi_{2}^{*}(t)=\frac{1}{n \varsigma H(t)}\left[\ln \frac{\delta}{\theta_3^* h^P \varsigma }+G_{i}(t)\right]. \end{array} $

The robust optimal reinsurance strategies are same as post-default case and the optimal value function is given in different situations.

(1) When $ D_{1} \leq D_{2} $, the value function is given by

$ V(t, x, y, l, 0)=\left\{\begin{aligned} -\frac{1}{n} \exp ^{-n (x+\varepsilon y)\bar H(t)+l\bar M(t)+\bar N_1(t)}, \; & 0 \leq t<t_{2}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) \bar H(t)+l\bar M(t)+\bar N_2(t)}, \; & t_{2} \leq t<t_{1}^{\prime}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) \bar H(t)+l\bar M(t)+\bar N_3(t)}, \; & t_{1}^{\prime} \leq t<T, \end{aligned}\right. $

(2) When $ D_{1}>D_{2} $, the value function is given by

$ V(t, x, y, l, 0)=\left\{\begin{aligned} -\frac{1}{n} \exp ^{-n (x+\varepsilon y) \bar H(t)+l\bar M(t)+\bar N_4(t)}, \; & 0 \leq t<t_{1}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) \bar H(t)+l\bar M(t)+\bar N_5(t)}, \; & t_{1} \leq t<t_{2}^{\prime}, \\ -\frac{1}{n} \exp ^{-n (x+\varepsilon y) \bar H(t)+l\bar M(t)+\bar N_3(t)}, \; & t_{2}^{\prime} \leq t<T, \end{aligned}\right. $

where $ H(t) $, $ \bar H(t) $, $ \bar M(t) $, $ G_{i}(t) $ and $ \bar N_{i}(t) $ are given by equations (4.13) and (4.36)-(4.39), respectively.

5 Numerical Analysis

In this section, we present some numerical examples to verify the theoretical results on the optimal strategy. Assuming that the claim sizes random variables $ X_i $ and $ Y_i $ are exponentially distributed with parameters $ \alpha_1 $ and $ \alpha_2 $, respectively, then $ \mu_1=1/\alpha_1, \; \mu_2=1/\alpha_2, \; \nu_1=1/\alpha_1^2 $ and $ \nu_2=1/\alpha_2^2 $. The basic parameters are given by $ \lambda=1, \; \lambda_1=3, \; \lambda_2=4, \; \beta_0=\beta_1=\beta_2=1, \; \theta=0.3, \; n=1.5, \; A=0.1, \; \varepsilon=0.1, \; r=0.04, \; \sigma=1, \; \alpha=2, \; \rho=0.3, \; T=10, \; t=0, \; h^p=0.003 $ and $ h^Q=0.02 $.

Based on the parameters we have set, it follows that $ U_1<U_2 $ and $ \theta U_2 < (n+\beta_0)U_3 $ always hold, so we set $ t_2 = T $ and the optimal reinsurance strategy $ (q_1^*(t), q_2^*(t)) = (\hat q_1(t), \hat q_2(t)) $. In Figure 1, the reinsurance proportion decreases with the increase of $ \lambda $. It means that the higher degree of dependence of the two insurance businesses, the greater potential risks of the insurance will face. From Figure 2, we can see that the reinsurance proportion decreases with the ambiguity-averse level $ \beta_0 $ increasing.

Figure 1 The effect of $\lambda$ on $q^*(t)$.

Figure 2 The effect of $\beta_0$ on $q^*(t)$.

Next, we analyze the effect of some parameters on the $ \pi^{*}_{1}(t) $ and $ \pi^{*}_{2}(t) $. In Figure 3, we know that $ \pi_1^*(t) $ decreases with the risk aversion coefficient n, which indicates that the insurer will pay less money to invest in stock market when n becomes larger. In Figure 4, with larger $ \beta_1 $, the insurer has less confidence in the reference model, and hence the less risky asset the insurer wishes to invest. In fact, the parameter $ \rho $ is the correlation coefficient between $ W_{1}(t) $ and $ \bar{W}_{2}(t) $. From Figure 5, with stronger positive correlation, the variance of the volatility becomes greater, which leads to that the AAI reduces the risk exposure. On the contrary, with stronger negative correlation, the AAI is willing to invest more wealth in the risky asset. In Figure 6, the wealth invested in the defaultable bond increases when the loss rate decreases. On the contrary, the insurer will invest more money in the defaultable bond with higher credit spread $ \delta $.

Figure 3 The effect of n on $\pi_1^*(t)$.

Figure 4 The effect of $\beta_1$ on $\pi_1^*(t)$.

Figure 5 The effect of $\rho$ on $\pi_1^*(t)$.

Figure 6 The effect of $\varsigma$ and $\delta$ on $\pi_2^*(t)$.
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