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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.

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)