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 $$dX_t=[1−|X_t|^2]^{1/2}γ(|X_t|) dB_t−g(|X_t|)X_t dt$$ 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 $h: \overline D\to \mathbb {R}$ Lipschitz on $\overline D$, 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 $$dX_t=h(X_t)^{1/2}σ(X_t)dB_t+b(X_t) dt$$ in $\overline D$.