## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 1 (2017), 404-468.

### On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries

Weining Kang and Kavita Ramanan

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 1 (2017), 404-468.

**Dates**

Received: December 2014

Revised: August 2016

First available in Project Euclid: 26 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1485421336

**Digital Object Identifier**

doi:10.1214/16-AOP1153

**Mathematical Reviews number (MathSciNet)**

MR3601653

**Zentralblatt MATH identifier**

06696273

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Secondary: 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

**Keywords**

Reflecting diffusions reflected Brownian motion submartingale problem martingale problem stochastic differential equations with reflection weak solutions Skorokhod problem extended Skorokhod problem nonsemimartingales diffusion approximations queueing networks

#### Citation

Kang, Weining; Ramanan, Kavita. On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries. Ann. Probab. 45 (2017), no. 1, 404--468. doi:10.1214/16-AOP1153. https://projecteuclid.org/euclid.aop/1485421336