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November 2010 The Skorohod oblique reflection problem in time-dependent domains
Kaj Nyström, Thomas Önskog
Ann. Probab. 38(6): 2170-2223 (November 2010). DOI: 10.1214/10-AOP538

Abstract

The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka [61] and further investigated by P.-L. Lions and A.-S. Sznitman [42] in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of càdlàg solutions (x, λ) to the Skorohod problem, with oblique reflection, for (D, Γ, w) assuming, in particular, that D is a time-dependent domain (Theorem 1.2). In addition, we prove that if w is continuous, then x is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.

Citation

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Kaj Nyström. Thomas Önskog. "The Skorohod oblique reflection problem in time-dependent domains." Ann. Probab. 38 (6) 2170 - 2223, November 2010. https://doi.org/10.1214/10-AOP538

Information

Published: November 2010
First available in Project Euclid: 24 September 2010

zbMATH: 1208.60077
MathSciNet: MR2683628
Digital Object Identifier: 10.1214/10-AOP538

Subjects:
Primary: 60J50 , 60J60

Keywords: Oblique reflection , Skorohod problem , Stochastic differential equations , time-dependent domain

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 6 • November 2010
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