Optimal controls for stochastic systems with singular noise (Q1820747)

The aim of the paper is the study of the existence of completely observed optimal controls for the stochastic system \[ (1)\quad dx_ t=f(t,x,u_ t)dt+g(t,x)db_ t\quad (0\leq t\leq 1),\quad x_ 0=0. \] The cost functional is \[ J(u,x)=\begin{cases} E(h(x)), &\text{ if x solves (1) for control u,} \\ +\infty, &\text{ otherwise.}\end{cases} \tag{2} \] Here \(g(t,z)\) is a singular \(n\times m\) matrix and \(h: (C[0,1])^ n\to R\) is a bounded non-negative continuous function. The results are obtained by nonstandard methods. The existence of a relaxed optimal control is proved. The existence of an ordinary optimal control is obtained under some additional convexity condition for the function f.