Uniqueness for diffusions degenerating at the boundary of a smooth bounded set
| dc.creator | DeBlassie, Dante | |
| dc.date | 2005-03-25 | |
| dc.date.accessioned | 2026-07-07T05:18:28Z | |
| dc.date.available | 2026-07-07T05:18:28Z | |
| dc.description | For continuous γ, g:[0,1]\to(0,\infty), 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 R^n. We introduce a new idea to show pathwise uniqueness holds when γand g are Lipschitz and \frac{g(1)}{γ^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with γ=\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}σ(X_t) dB_t+b(X_t) dt in \barD. | |
| dc.description | Published at http://dx.doi.org/10.1214/009117904000000810 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org) | |
| dc.identifier | https://arxiv.org/abs/math/0503590 | |
| dc.identifier | http://arxiv.org/abs/math/0503590 | |
| dc.identifier | Annals of Probability 2004, Vol. 32, No. 4, 3167-3190 | |
| dc.identifier | doi:10.1214/009117904000000810 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74671 | |
| dc.subject | Probability | |
| dc.subject | 60H10, 60J60 (Primary) | |
| dc.title | Uniqueness for diffusions degenerating at the boundary of a smooth bounded set | |
| dc.type | text |