Abstract
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.
Highlights
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, risky investment, and bond investment. e insurer can 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
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, 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
Default 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
Summary
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 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. 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. 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. 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.
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