Abstract
This paper investigates the excess-of-loss reinsurance and investment problem for a compound Poisson jump-diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV) model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton-Jacobi-Bellman (HJB) equation is established. Moreover, we show that a closed-form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motionsW(t)andW1(t). A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealthxand decreasing with the volatility rate of risk asset price. However, the optimal value functionV(t;x;s)is increasing with the appreciation rateμof risk asset.
Highlights
By means of investment and reinsurance, insurers can protect themselves against potentially large losses or ensure their earnings remain relatively stable
Let us give the following exponential utility function of an risk aversion insurer: U (x) = − 1 e−qx, q > 0. q. This utility function has a constant absolute risk aversion parameter q and is the only utility function under the principle of “zero utility” giving a fair premium that is independent of the level of reserves of insurers
From Theorem 1, we find that the optimal investment strategy is a function of (μ − r), q, x, ksβ, r, and t, which fully reflects the influence of various factors on the investment strategy. x is the initial wealth, ksβ represents the volatility of risk asset price, and μ − r is the profit of risk asset appreciation higher than risk-free interest rate
Summary
By means of investment and reinsurance, insurers can protect themselves against potentially large losses or ensure their earnings remain relatively stable. Many optimal investment and reinsurance problems have arisen in insurance risk management and have been extensively studied in the literature. The excess-of-loss reinsurance is a tool commonly employed in risk management in the recent thirty years. Tapiero and Zuckerman [2] gave the optimum excess-loss reinsurance under a dynamic framework. Cao and Xu [4] investigated both proportional and excess-of-loss reinsurance under investment gains. Gu et al [5] investigated optimal control of excess-of-loss reinsurance and investment for insurers under a constant elasticity of variance model but without compound Poisson jump in their research. Zhao et al [6] studied optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model
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