## Optimal control of capital injections by reinsurance in a diffusion approximation.(English)Zbl 1183.91069

Summary: In this paper we consider a diffusion approximation to a classical risk process, where the claims are reinsured by some reinsurance with deductible $$b \in [0,\tilde b]$$, where $$b = \tilde b$$ means “no reinsurance” and $$b = 0$$ means “full reinsurance”. The cedent can choose an adapted reinsurance strategy $$(b_t )_{t \geq 0}$$, i.e. the deductible can be changed continuously. In addition, the cedent has to inject fresh capital in order to keep the surplus positive. The problem is to minimize the expected discounted cost over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach. Some examples illustrate the method.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 93E20 Optimal stochastic control 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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### References:

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