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Probability and Math Finance Seminar
Michael Harrison, Stanford UniversityRecurrence Classification of Semimartingale Reflecting Brownian Motions Abstract
Let be an -dimensional Brownian motion confined to the non-negative orthant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for multi-station stochastic processing networks. The parameters of are a drift vector and covariance matrix that characterize movement away from the boundary, plus an matrix R whose columns specify ``directions of reflection'' - more aptly called ``directions of displacement'' - from the orthant's boundary faces. The matrix is assumed to be completely-, which is necessary and sufficient for to be a semimartingale. This is the case of primary interest in applications. For dimension , a simple condition is known to be necessary and sufficient for positive recurrence of ; it involves and but not . The obvious analog of that condition is necessary but not sufficient in three and higher dimensions, where fundamentally new phenomena arise. This was established in a little-known pair of papers by El Kharroubi et al. (2000, 2002). Building on seminal earlier work by Bernard and El Kharroubi (1991) and Dupuis and Williams (1994), those authors also proved important positive results for dimension , but they left several questions unresolved. In this talk I will first review the work described above. Extrapolating from it in an obvious way, I will conjecture that a certain multi-part condition is necessary and sufficient for positive recurrence in dimension . Recent progress toward verification of the conjecture will be described, as will remaining open problems. Based on joint work with Jim Dai. MONDAY, March 17, 2008 |