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Robust optimal proportional reinsurance and investment strategy for an insurer with defaultable risks and jumps. (English) Zbl 1410.91295
Summary: In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with default risks and jumps for an insurer whose objective is to maximize the expected exponential utility of terminal wealth. The surplus process of the insurer is assumed to follow a Brownian motion with drift (which is an approximation of the classical compound Poisson risk model). Assume that the insurer is allowed to purchase proportional reinsurance and invest in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset whose price process is governed by a jump-diffusion model. Using stochastic control approach, we establish the robust Hamilton-Jacobi-Bellman equations for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed-form expressions of the optimal strategies and the corresponding value functions in both cases. Finally, we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control 60J75 Jump processes (MSC2010)
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