Abstract

We consider an optimal control problem for a onedimensional 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 pathwisedeterministic measuredriven 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 pathwisedeterministic 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.

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