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On controlled diffusion processes in a bounded domain. (English. Russian original) Zbl 0869.60047
Theory Probab. Appl. 40, No. 3, 533-538 (1995); translation from Teor. Veroyatn. Primen. 40, No. 3, 632-638 (1993).
It is known that controlled diffusion processes are described by the stochastic differential equations $dx_t= \sigma (x_t,\alpha_t) dw_t+ b(x_t, \alpha_t)dt, \quad t\geq 0, \quad x_0=x, \tag{1}$ where $$\alpha_t$$ is a random process with values in a set $$A$$ measurable in an appropriate way. By means of $$\alpha_t$$, a control is established whose property is estimated by the expression of the form $$E\int^\tau_0 e^{-t} f(x_t, \alpha_t)dt$$, where $$\tau$$ is the first exit time of a process $$x_t$$ from a domain $$D\subset E_d$$. The paper studies a situation similar to that described by N. V. Krylov [Teor. Veroyatn. Primen. 31, No. 4, 685-709 (1986; Zbl 0619.93069)] for the whole space $$E_d$$. By the solution of equation (1) we understand the weak solution; weakened controls are considered. The existence of an optimal control with “good” properties is demonstrated. The proof is based on the results of N. V. Krylov (loc. cit.) and the author and N. V. Krylov [Theory Probab. Appl. 33, No. 1, 1-10 (1988); translation from Teor. Veroyatn. Primen. 33, No. 1, 3-13 (1988; Zbl 0671.60052)].
Reviewer: L.A.Alyushina
##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G35 Signal detection and filtering (aspects of stochastic processes) 93E20 Optimal stochastic control