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This article is cited in 10 papers
On Stefan's problem and optimal stopping rules for Markov processes
B. I. Grigelionis, A. N. Shiryaev
Moscow
Abstract:
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).
Received: 25.04.1966
Citation:
B. I. Grigelionis, A. N. Shiryaev, “On Stefan's problem and optimal stopping rules for Markov processes”, Teor. Veroyatnost. i Primenen., 11:4 (1966), 612–631
Citation in format AMSBIB:
\Bibitem{1}
\by B.~I.~Grigelionis, A.~N.~Shiryaev
\paper On Stefan's problem and optimal stopping rules for Markov processes
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 4
\pages 612--631
\mathnet{http://mi.mathnet.ru/tvp662}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=216709}
\zmath{http://www.zentralblatt-math.org/zmath/search/?an=Zbl 0178.53303}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 4
\pages 541--558
\crossref{http://dx.doi.org/10.1137/1111060}
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English version:
Theory of Probability and its Applications, 1966, 11:4, 541–558
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 ; N. V. Krylov, “Control of Markov processes and $W$-spaces”, Math. USSR-Izv., 5:1 (1971), 233–266 -
Н. В. Крылов, “О единственности решения уравнения Беллмана”, Изв. АН СССР. Сер. матем., 35:6 (1971), 1377–1388 ; N. V. Krylov, “On uniqueness of the solution of Bellman's equation”, Math. USSR-Izv., 5:6 (1971), 1387–1398
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Е. Б. Фрид, “О полурегулярности граничных точек для нелинейных уравнений”, Матем. сб., 94(136):4(8) (1974), 516–539 ; E. B. Frid, “On the semiregularity of boundary points for nonlinear equations”, Math. USSR-Sb., 23:4 (1974), 483–507
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Г. Каллианпур, О. А. Олейник, “О задачах со свободной границей, возникающих в теории вероятностей
(теоремы единственности)”, УМН, 51:6(312) (1996), 205–206
 ; G. Kallianpur, O. A. Oleinik, “On free boundary problems arising in probability theory (uniqueness theorems)”, Russian Math. Surveys, 51:6 (1996), 1203–1205  -
Е. В. Ягнятинский, “Одна задача оптимального последовательного инвестирования”, УМН, 52:4(316) (1997), 193–194 ; E. V. Yagnyatinskii, “A problem of optimal sequential investing”, Russian Math. Surveys, 52:4 (1997), 850–851
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Del Moral P., Guionnet A., “Large deviations for interacting particle systems: Applications to non–linear filtering”, STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 78:1 (1998), 69–95
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Dayanik S., Karatzas L., “On the optimal stopping problem for one–dimensional diffusions”, STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 107:2 (2003), 173–212
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Lototsky S.V., “Wiener chaos and nonlinear filtering”, APPLIED MATHEMATICS AND OPTIMIZATION, 54:3 (2006), 265–291
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Dayanik S., “Optimal stopping of linear diffusions with random discounting”, MATHEMATICS OF OPERATIONS RESEARCH, 33:3 (2008), 645–661
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D. V. Belomestny, L. Rüschendorf, M. A. Urusov, “Optimal Stopping of Integral Functionals and a “No-Loss” Free Boundary Formulation”, ТВП, 54:1 (2009), 80–96 ; Theory Probab. Appl., 54:1 (2010), 14–28
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