The Annals of Probability

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

Dante DeBlassie

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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 gc, 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 $\overline D$.

Article information

Source
Ann. Probab., Volume 32, Number 4 (2004), 3167-3190.

Dates
First available in Project Euclid: 8 February 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1107883350

Digital Object Identifier
doi:10.1214/009117904000000810

Mathematical Reviews number (MathSciNet)
MR2094442

Zentralblatt MATH identifier
1071.60043

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

Keywords
Degenerate stochastic differential equations diffusions non-Lipschitz coefficients pathwise uniqueness weak uniqueness submartingale problem

Citation

DeBlassie, Dante. Uniqueness for diffusions degenerating at the boundary of a smooth bounded set. Ann. Probab. 32 (2004), no. 4, 3167--3190. doi:10.1214/009117904000000810. https://projecteuclid.org/euclid.aop/1107883350


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