For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.
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References
A. D. Wentzell, “Semigroups of operators corresponding to a generalized differential operator of the second order,” Dokl. Akad. Nauk SSSR, 111, No. 2, 269–272 (1956).
A. D. Wentzell, “On boundary condition for multidimensional diffusion processes,” Teor. Ver. Primen., 4, No. 2, 172–185 (1959).
A. V. Skorokhod, “Stochastic equations for diffusion processes with boundaries. I,” Teor. Ver. Primen., 6, Issue 3, 287–298 (1961).
S. Watanabe, “Construction of diffusion processes with Wentzell’s boundary conditions by means of Poisson point process of excursions,” in: Proceedings of the Third Soviet–Japanese Symposium on Probability Theory, Fan, Tashkent (1975), pp. 311–345.
S. V. Anulova, “On stochastic differential equations with boundary conditions in a half space,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 3, 491–508 (1981).
S. V. Anulova, “On processes with generating Lévy operator in a half space,” Izv. Akad. Nauk SSSR, Ser. Mat., 42, No. 4, 708–750 (1978).
R. Mikulyavichyus, “On the existence of solutions of the martingale problem,” Lit. Mat. Sb., 17, No. 4, 149–168 (1977).
R. Mikulyavichyus, “On the uniqueness of solutions of the martingale problem,” Lit. Mat. Sb., 18, No. 2, 63–73 (1978).
T. Komatsu, “Markov processes associated with certain integro-differential operators,” Osaka J. Math., 10, No. 2, 271–303 (1973).
K. Sato and T. Ueno, “Multi-dimensional diffusion and the Markov process on the boundary,” J. Math. Kyoto Univ., 4, 529–605 (1965).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam (1981).
V. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations [in Russian], Naukova Dumka, Kiev (1968).
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer Berlin (1979).
P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).
P. E. Protter, Stochastic Integration and Differential Equations, Springer (2004).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 9, pp. 1241–1256, September, 2011.
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Pilipenko, A.Y. On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary. Ukr Math J 63, 1415–1432 (2012). https://doi.org/10.1007/s11253-012-0588-2
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DOI: https://doi.org/10.1007/s11253-012-0588-2