Uniqueness for diffusions degenerating at the boundary of a smooth bounded set

Abstract

For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise uniqueness holds when \gamma and g are Lipschitz and \frac{g(1)}{\gamma^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with \gamma=\sqrt2 and g\equiv c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C^3 boundary and suppose h:\barD\to R Lipschitz on \barD, as well as C^2 on a neighborhood of \partial D with Lipschitz second partials there. Also assume h>0 on D, h=0 on \partial D and |\nabla h|>0 on \partial D. An example of such a function is h(x)=d(x,\partial D). We give conditions which ensure pathwise uniqueness holds for dX_t=h(X_t)^{1/2}\sigma(X_t) dB_t+b(X_t) dt in \barD.