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October 2004 Uniqueness for diffusions degenerating at the boundary of a smooth bounded set
Dante DeBlassie
Ann. Probab. 32(4): 3167-3190 (October 2004). DOI: 10.1214/009117904000000810


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 $\overline D$.


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Dante DeBlassie. "Uniqueness for diffusions degenerating at the boundary of a smooth bounded set." Ann. Probab. 32 (4) 3167 - 3190, October 2004.


Published: October 2004
First available in Project Euclid: 8 February 2005

zbMATH: 1071.60043
MathSciNet: MR2094442
Digital Object Identifier: 10.1214/009117904000000810

Primary: 60H10 , 60J60

Keywords: Degenerate stochastic differential equations , Diffusions , non-Lipschitz coefficients , Pathwise uniqueness , submartingale problem , Weak uniqueness

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 4 • October 2004
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