The Annals of Probability

The Skorohod oblique reflection problem in time-dependent domains

Kaj Nyström and Thomas Önskog

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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.

Article information

Source
Ann. Probab., Volume 38, Number 6 (2010), 2170-2223.

Dates
First available in Project Euclid: 24 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1285334204

Digital Object Identifier
doi:10.1214/10-AOP538

Mathematical Reviews number (MathSciNet)
MR2683628

Zentralblatt MATH identifier
1208.60077

Subjects
Primary: 60J50: Boundary theory 60J60: Diffusion processes [See also 58J65]

Keywords
Skorohod problem oblique reflection time-dependent domain stochastic differential equations

Citation

Nyström, Kaj; Önskog, Thomas. The Skorohod oblique reflection problem in time-dependent domains. Ann. Probab. 38 (2010), no. 6, 2170--2223. doi:10.1214/10-AOP538. https://projecteuclid.org/euclid.aop/1285334204


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