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Optimal dynamic reinsurance policies for large insurance portfolios. (English) Zbl 1066.91052
The authors consider a problem of minimization of the ruin probability for an insurance company whose surplus is relatively large compared to the size of individual claim and is modelled by a diffusion process. It is assumed that without reinsurance the dynamics of the insurance company surplus is described by the equation $dR_t=\mu dt+\sigma dw^{(1)}_t,\quad R_0=x,$ and for the proportional reinsurance the corresponding diffusion approximation becomes $dR_t=(\mu-(1-a)\lambda)dt+\sigma dw^{(1)}_t,$ where $$0\leq a\leq 1$$ is called risk exposure, $$\lambda\geq \mu$$. In addition, it is assumed that all of the surplus is invested in a stock market instrument, whose price is governed by the classical Black-Scholes dynamics $dP_t =rP_tdt+\sigma_pP_tdw^{(2)}_t.$ Stochastic optimal control theory is used to determine the optimal reinsurance policy which minimizes the ruin probability of the cedent. It is demonstrated that the optimal policy of the cedent depend significantly on the nature of the investment available, in particular on its volatility. The economic analysis of all possible relations between coefficients is given and numerical results are presented.

##### MSC:
 91G10 Portfolio theory 91B30 Risk theory, insurance (MSC2010) 93E11 Filtering in stochastic control theory 93E20 Optimal stochastic control 60H30 Applications of stochastic analysis (to PDEs, etc.)
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