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

• 成果类型: Article
• 署名作者: DeBlassie, D
• 署名单位: Texas A&M University System; Texas A&M University College Station
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000810
• 发表日期: 2004
• 页码: 3167-3190
• 关键词: sde

摘要

For continuous gamma, g:[0, 1] --> (0, infinity), consider the degenerate stochastic differential equation dX(t) = [1 - \X-t \(2)](1/2)gamma(\X-t\)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 gamma and g are Lipschitz and g(1)/gamma(2)(1) > root2 -1 . When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with gamma = root2 and g equivalent to 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: (D) over bar --> R Lipschitz on (D) over bar, as well as C-2 on a neighborhood of thetaD with Lipschitz second partials there. Also assume h > 0 on D, h = 0 on thetaD and \delh\ > 0 on thetaD. An example of such a function is h(x) = d(x, thetaD). We give conditions which ensure pathwise uniqueness holds for dX(t) = h(X-t)(1/2)sigma(X-t)dB(t) + b(X-t)dt in (D) over bar.