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Russia Ufa
Section Mathematics
Title On deterministic approach to solution of stochastic optimal control problem with controlled diffusion
Author(-s) Ismagilov N.S.a
Affiliations Ufa State Aviation Technical Universitya
Abstract We consider an optimal control problem for a one-dimensional process driven by stochastic differential equation, which has both drift and diffusion coefficients controlled, diffusion being linear in control $$dx(t) = b(t,x(t),u(t))\,dt + \sigma(t,x(t))u(t)\,dW(t), \quad x(0) = x_0,$$ where $x(t)$ is the state variable, $u(t)$ is the control variable and $W(t)$ is the Wiener process. We prove a theorem which gives a structure of solution for the considered differential equation as a superposition of functions $x(t) = \Phi(t,u(t)W(t) + y(t))$, where $\Phi(t,v)$ is the known function, which is completely determined by the diffusion coefficient $\sigma(t,x)$ and is independent of control, and $y(t)$ is the solution to the pathwise-deterministic measure-driven differential equation $$dy(t) = B(t,y(t),u(t))\,dt - W(t)\,du(t).$$ The revealed structure of the solution enables us to consider a new pathwise-deterministic impulsive optimal control problem with the state variable $y(t)$ which is equivalent to the original stochastic optimal control problem. Pathwise problems may have anticipative solutions, so we propose a method that makes it possible to build nonanticipative optimal solutions. The basic idea of the method is to modify cost functional in new pathwise problem with special integral term, which guarantees nonanticipativity of solutions.
Keywords stochastic optimal control, stochastic differential equations, deterministic approach, pathwise optimization, optimal impulsive control
UDC 519.21, 517.977
MSC 93E20, 49K45, 60H30, 49N25
DOI 10.20537/vm140202
Received 29 October 2013
Language Russian
Citation Ismagilov N.S. On deterministic approach to solution of stochastic optimal control problem with controlled diffusion, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 29-42.
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