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January 2017 On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries
Weining Kang, Kavita Ramanan
Ann. Probab. 45(1): 404-468 (January 2017). DOI: 10.1214/16-AOP1153

Abstract

Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection (SDER) and the so-called submartingale problem. We consider a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise $\mathcal{C}^{2}$ boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding SDER. The main step involves showing existence of a weak solution to the SDER given a solution to the submartingale problem. This generalizes the classical construction, due to Stroock and Varadhan, of a weak solution to an (unconstrained) stochastic differential equation, but requires a completely different approach to deal with the geometry of the domain and directions of reflection and properly identify the local time on the boundary, in the presence of multi-valued directions of reflection at nonsmooth parts of the boundary. In particular, our proof entails the construction of classes of test functions that satisfy certain oblique derivative boundary conditions, which may be of independent interest. Other ingredients of the proof that are used to identify the constraining or local time process include certain random linear functionals, suitably constructed exponential martingales and a dual representation of the cone of directions of reflection. As a corollary of our result, under suitable assumptions, we also establish an equivalence between well-posedness of both the SDER and submartingale formulations and well-posedness of the constrained martingale problem, which is another framework for defining (semimartingale) reflected diffusions. Many of our intermediate results are also valid for reflected diffusions that are not necessarily semimartingales, and are used in a companion paper [Equivalence of stochastic equations and the submartingale problem for nonsemimartingale reflected diffusions. Preprint] to extend the equivalence result to a class of nonsemimartingale reflected diffusions.

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Weining Kang. Kavita Ramanan. "On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries." Ann. Probab. 45 (1) 404 - 468, January 2017. https://doi.org/10.1214/16-AOP1153

Information

Received: 1 December 2014; Revised: 1 August 2016; Published: January 2017
First available in Project Euclid: 26 January 2017

zbMATH: 06696273
MathSciNet: MR3601653
Digital Object Identifier: 10.1214/16-AOP1153

Subjects:
Primary: 60H10, 60J60, 60J65
Secondary: 90B15, 90B22

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 1 • January 2017
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