The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Abstract

The Skorohod oblique reflection problem for (D, Γ, w) (D a general domain in ℝ d , Γ(x),x∈∂D, a convex cone of directions of reflection,w a function inD(ℝ+,ℝ d )) is considered. It is first proved, under a condition on (D, Γ), corresponding to Γ(x) not being simultaneously too large and too much skewed with respect to ∂D, that given a sequence {wn} of functions converging in the Skorohod topology tow, any sequence {(xn, ϕn)} of solutions to the Skorohod problem for (D, Γ, wn) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, Γ, w). Next it is shown that if (D, Γ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, Γ, w) exist for everyw∈D(ℝ+,ℝ d ) with small enough jump size. The requirement is met in the case when ∂D is piecewiseC b 1 , Γ is generated by continuous vector fields on the faces ofD and Γ(x) makes and angle (in a suitable sense) of less than π/2 with the cone of inward normals atD, for everyx∈∂D. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.