On the optimal control by means of a random time substitution in a continuous Markov process

Let $X$ be a homogeneous Markov Feller process with continuous paths in a compact $E$. For the process $X^u$ obtained from $X$ by means of a random time substitution connected with the additive functional (1), we prove the existence of a continuous optimal control $u^*(x)$ (4) that minimizes the risk $R^u(x)$ (2). Further, we show that the optimal risk $R^*(x)$ is the only continuous solution of the equation (3), where $A$ is the weak infinitesimal operator of $X$. Under some assumptions we obtain an equation for a lower bound $r(x)$ of the optimal risk $R^*(x)$.