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Optimal risk control for a large corporation in the presence of returns on investments. (English) Zbl 1049.93090
Recently there has been a renewed interest in diffusion models for corporations with controllable risk exposure and dividends distribution. For more details see, for example, a survey by M. Taksar [Math. Methods Oper. Res. 51, No. 1, 1–42 (2000; Zbl 0947.91043)]. In this paper a firm valuation problem is considered for a company that has control on the dividend payment stream and its risk. The company invests its reserve in a financial market which may or may not contain an element of risk. The company chooses a dividend payment policy and the value of the company is associated with the expected present value of the net dividend distributions. One of the examples could be a large corporation such as an insurance company whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. The diffusion coefficient is interpreted as risk exposure while drift is interpreted as potential profit. The company invests its reserve in a financial asset whose price evolves as a geometric Brownian motion with mean rate $$r>0$$ and diffusion constant $$\sigma_P\geq 0$$. The problem is to find a policy consisting of risk control and dividend payment scheme which maximizes the expected total discounted dividends paid out until the time of bankruptcy. Theory of controlled diffusions (see, for example, W. H. Fleming and R. W. Rishel [Deterministic and stochastic optimal control. Berlin: Springer-Verlag (1975; Zbl 0323.49001)]) is applied to solve the problem. It is shown that if the discount rate $$c$$ is less than $$r$$, then the optimal return function is infinite. If $$r=c$$ the return function is finite for all $$x<\infty$$, but no optimal policy exists. If $$r<c$$, then there is a finite level $$u_1>0$$ such that the optimal action is to distribute all reserve exceeding $$u_1$$ as dividends. Furthermore there exists a constant $$x_0$$ with $$x_0<u_1$$ such that the risk exposure increases monotonically on $$(0,x_0)$$ from 0 to the maximum possible value.

##### MSC:
 93E20 Optimal stochastic control 91B30 Risk theory, insurance (MSC2010) 49L20 Dynamic programming in optimal control and differential games 62P05 Applications of statistics to actuarial sciences and financial mathematics
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