Optimal excess-of-loss reinsurance and investment problem with thinning dependent risks under Heston model

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Abstract

This paper studies an optimal excess-of-loss reinsurance and investment problem with thinning dependent risks. Assume that the insurer’s wealth process is described by a risk model with two dependent classes of insurance business, and the insurer is allowed to purchase excess-of-loss reinsurance from the reinsurer and invest in a risk-free asset and a risky asset whose price follows Heston model. Our aim is to seek the optimal excess-of-loss reinsurance and investment strategy under the criterion of maximizing the expected exponential utility of the terminal wealth. Applying Legendre transform along with the stochastic control theory, we obtain the explicit expressions of the optimal excess-of-loss reinsurance and investment strategy. Finally, we give some numerical examples to illustrate our results.

Introduction

In recent years, the optimal excess-of-loss reinsurance and investment problems in insurance risk management have been paid great attention by many scholars. For example, Asmussen et al. [1] firstly studied the optimal dividend problem under the control of excess-of-loss reinsurance. Gu et al. [2] derived optimal excess-of-loss reinsurance and investment strategies under the CEV model, which is similar to that of Asmussen et al. [1]. They showed that excess-of-loss reinsurance is more profitable than the proportional reinsurance. Under the criterion of maximizing the expected exponential utility of the insurer’s terminal wealth, Zhao et al. [3] obtained the optimal excess-of-loss reinsurance and investment strategy under Heston model. A and Li [4] generalized the results in Zhao et al. [3] and optimized the delayed problem of excess-of-loss reinsurance and investment. Li et al. [5] took ambiguity into account and analyzed a robust optimal problem of excess-of-loss reinsurance and investment in a model with jumps. Li et al. [6] studied an insurer’s reinsurance-investment problem under a mean–variance criterion.

Although the optimal excess-of-loss reinsurance and investment problem has been widely investigated, only few papers deal with the problem in relation to dependent risks. Recently, more and more natural disaster and manmade disaster frequently occurred in the whole world which brought great damage to the safety of life and property for people. In practice, the insurance businesses are usually dependent through some way. Therefore, it is necessary to consider the dependent risk model in the actuarial literature. Under the criterion of maximizing the expected utility of terminal wealth, Liang and Yuen [7] and Yuen et al. [8] considered the optimal proportional reinsurance strategies in a risk model with two or more dependent classes of insurance business, where the claim number processes are correlated through a common shock component. Liang et al. [9] and Bi et al. [10] studied the optimal investment and reinsurance problems with dependence under the criterion of mean–variance. For other research about dependent risks, we refer readers to [11], [12], [13], [14] and the references therein.

Besides the risk common shock dependent model mentioned above, the different classes of insurance business may be dependent through other ways. A typical example is that a traffic accident (for example, fire accidents, car accidents and aviation accidents and so on) may cause property loss or medical claims or death claims. If a traffic accident causes medical claims with a certain probability p, the medical claim number process is a p-thinning process (the so-called p-thinning process roughly speaking is that, assume the occurrence of event E forms a renewal process N(t), if the occurrence of each event E is recorded by probability p, which forms a random record process Np(t), then we call Np(t) is a p-thinning process of N(t)) of the claim number process of car insurance, which can be called p-thinning-dependence. Wang and Yuen [15] presented a risk model with n (2) dependent classes of insurance business in which there exists certain correlation between the n claim-number processes due to the so-called thinning-dependence structure. To the best of our knowledge, there are very few results concerning the optimal excess-of-loss reinsurance and investment problem in relation to thinning dependent risks. In this paper, we consider an optimal excess-of-loss reinsurance and investment problem in a risk model with two dependent classes of insurance business, in which the second claim number process is the p-thinning process of the first claim process. The insurer is assumed to invest in a risk-free asset and a risky asset whose price process follows Heston model. To study more practical financial market, Heston [16] assumed that the volatility of the risky asset was driven by a Cox–Ingersoll–Ross (CIR) process, this model has some computational and empirical advantages. Actually, the Heston model is classical and very popular for option pricing, and has been recognized as an important feature for asset price models. Meanwhile, the Heston model can explain many well-known empirical findings, such as the volatility smile, the volatility clustering, and the heavy-tailed nature of return distributions.

Our aim is to maximize the expected exponential utility of terminal wealth. Firstly, applying stochastic control theory, we establish the Hamilton–Jacobi–Bellman (HJB) equation for the value function. However, the HJB equation in this paper is a non-linear second order partial differential equation (PDE), which is difficult to be simplified or solved directly in general. Motived by Chang and Chang [17] and Zhang and Zhao [18], we adopt the Legendre transform-dual technique to change this HJB equation into its dual one, as a result, the form of its solution is easy to be conjectured. We apply variable change technique to obtain the closed-form expression of optimal strategy. This paper has the main highlights as follows: (i) dependent risks are considered in the optimal excess-of-loss reinsurance-investment problem; (ii) Legendre transform method is used to deal with the optimal control problem.

The rest of this paper is organized as follows. The formulation of our model is presented in Section 2. In Section 3, we discuss the optimal excess-of-loss optimal reinsurance strategy. Section 4 derives the optimal investment strategy by using Legendre transform method. In Section 5, numerical examples are carried out to illustrate our results in this paper. Finally, we conclude the paper in Section 6.

Section snippets

Model formulation

We start with a filtered complete probability space (Ω,F,{Ft}t[0,T],P) , where T represents the terminal time which is a positive finite constant, Ft 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. In addition, suppose that trading takes place continuously and involves no taxes or transaction costs, and that all securities are infinitely divisible.

Optimal reinsurance strategy

The purpose of this section is to find the optimal control strategy (m1,m2,π) for the problem (10). We hope to find a nice enough function satisfying (14), (11). Suppose that the insurer has an exponential utility function as follows u(x)=1vevx,where v>0 is a constant absolute risk aversion parameter.

To solve (14), we conjecture a solution in the following form V(t,x,l)=1vexp{v[xer(Tt)+G(t,l)]},with G(T,l)=0.

Note that, the form of the value function V(t,x,l) given by (16) is only used

Optimal investment strategy

In this section, we aim to solve Eq. (36). In what follows, applying Legendre transform-dual theory, we firstly convert (36) into its dual one and then get a linear PDE, which is easy to be solved.

Definition 4.1

Let ϖ:RnR be a convex function. Legendre transform can be defined as follows: H(z)=maxx{zTxϖ(x)},then the function H(z) is called Legendre dual function of ϖ(x) (see Chang and Chang [17], Gao [25] and Xiao et al. [26]).

Remark 4

If ϖ(x) is strictly convex, the maximum in (38) will be attained at just one

Numerical analysis

In this section, we investigate the effects of parameters on the optimal strategies and present some numerical simulations to illustrate the effects. Here we only provide the analysis for the case of ξ1val1. For drawing convenience, assume that the claim sizes Xi and Yi follow uniform distribution U(0,1). Throughout this section, unless otherwise stated, the basic parameters are given by v=1.5,t=5,T=10,ρ=0.5,ξ1=2,ξ2=1,r=0.05,λ=3,p=0.2,σ=0.16,β=2,δ=0.3,α=1.5. We can draw the following

Conclusions

For the optimal excess-of-loss reinsurance and investment problems in insurance, few papers consider the control systems with dependent risks, while this paper investigates an optimal excess-of-loss reinsurance-investment problem in relation to thinning dependent risks under Heston model, and we obtain the closed-form expressions of the optimal strategy. To make the optimal control problem closer to reality, we furthermore consider some possible extensions of this paper. For example, we can

Acknowledgments

The authors gratefully thank the anonymous referees for their valuable suggestions.

References (28)

This work was supported by NNSF of China (No. 11871275; No. 11371194).

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