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

Abstract

For continuous γ, g:[0,1]→(0,∞), consider the degenerate stochastic differential equation $$d{X}_t=[1-|X_t{|}^2{]}^{1/2}\gamma (|X_t|)\hspace{0.17em}d{B}_t-g(|X_t|)X_t\hspace{0.17em}dt$$ in the closed unit ball of ℝⁿ. We introduce a new idea to show pathwise uniqueness holds when γ and g are Lipschitz and $\frac{g(1)}{{\gamma }^2(1)}>\sqrt{2}-1$. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131–149] with $\gamma =\sqrt{2}$ and g≡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 C³ boundary and suppose $h:\bar{D}\to \mathbb{R}$ Lipschitz on $\bar{D}$, as well as C² on a neighborhood of ∂D with Lipschitz second partials there. Also assume h>0 on D, h=0 on ∂D and |∇h|>0 on ∂D. An example of such a function is h(x)=d(x,∂D). We give conditions which ensure pathwise uniqueness holds for $$d{X}_t=h(X_t{)}^{1/2}\sigma (X_t)d{B}_t+b(X_t)\hspace{0.17em}dt$$ in $\bar{D}$.