On Stefan's problem and optimal stopping rules for Markov processes

Let $X=\{x_i,\zeta,\mathscr M_i,\mathbf P_x\}$ be a homogeneous Markov process with the phase space $E\subseteq R^n$. Let us denote $\tilde s(x)=\sup\limits_{\tau\in\mathfrak M}\mathbf M_xg(x_\tau)$ where $\mathfrak M$ is the class of Markov stopping moments. The purpose of this article is to find those conditions under which the finding of the optimal stopping moment $\widetilde\tau$ and the “cost” $\widetilde s(x)$ is equivalent to the solution of generalized Stefan's problem (5).