Geodesic
On Stefan's problem and optimal stopping rules for Markov processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 4, pp. 612-631

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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). 
@article{TVP_1966_11_4_a2,
     author = {B. I. Grigelionis and A. N. Shiryaev},
     title = {On {Stefan's} problem and optimal stopping rules for {Markov} processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {612--631},
     publisher = {mathdoc},
     volume = {11},
     number = {4},
     year = {1966},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1966_11_4_a2/}
}
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B. I. Grigelionis; A. N. Shiryaev. On Stefan's problem and optimal stopping rules for Markov processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 4, pp. 612-631. http://geodesic.mathdoc.fr/item/TVP_1966_11_4_a2/