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Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. (English) Zbl 0958.91026
The problem of optimal dynamic risk control/dividends distribution of a financial corporation has recently gained a lot of attention in Mathematical Finance. In some papers dividend optimisation was considered. In other papers only risk was controlled, with the dividend distribution scheme being exogenously fixed or with criteria related to other than dividends optimization. The most typical example of a financial corporation in this type of problems is an insurance company. By and large the risk control within an insurance company is maintained by a process called reinsurance. The reinsurance company receives from the primary insurance carrier (the cedent) a certain part of all premiums. In return it is obliged to pay a certain part of each claim according to an a priori established rule.
In this paper one of the most common cases in the reinsurance industry, of the excess-of-loss reinsurance is treated. Within this scheme the cedent pays all of the claim up to a fixed amount \(a\) (the retension level), while all in excess of \(a\) is paid by the reinsurance company. The problem, which is faced by the cedent is how to chose dynamically in an optimal way the retension level \(a\) and the dividend distribution policy. In this paper the diffusion approximation for this optimal control problem is used. Two situations are considered:
A) The rate of dividend pay-out are unrestricted and in this case mathematically the problem becomes a mixed singular-regular control problem for diffusion processes. Its analytical part is related to a free boundary (Stefan) problem for a linear second order differential equation.
B) The rate of dividend pay-out is bounded by some positive constant \(M<\infty\), in which case the problem becomes a regular control problem.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J60 Diffusion processes
93E20 Optimal stochastic control
35K20 Initial-boundary value problems for second-order parabolic equations
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
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