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Optimal investment and proportional reinsurance under no short-selling and no borrowing. (English) Zbl 1243.93139
Summary: Insurance companies resort to investment and reinsurance, among other options, to manage their reserves. This article addresses the problem of optimal investment and reinsurance when no short-selling and no borrowing allowed. More specifically, we assume that the risk process of the insurance company is a compound Poisson process perturbed by a standard Brownian motion and that the risk can be reduced through a proportional reinsurance. In addition, the surplus can be invested in the financial market such that the portfolio will consist, for simplicity, of one risky asset and one risk-free asset. Our goal is to find the optimal investment and reinsurance policy which can maximize the expected exponential utility of the terminal wealth. In the case of no short-selling, we find the closed form of value function as well as the optimal investment-reinsurance policy. In the case when neither short-selling nor borrowing allowed, the resulting HJB equation is difficult to solve analytically, and hence we provide a numerical solution through Markov chain approximation techniques.

MSC:
93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91B30 Risk theory, insurance (MSC2010)
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