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Probability and Math Finance Seminar

Michael Harrison, Stanford University

Recurrence Classification of Semimartingale Reflecting Brownian Motions

Abstract

 

Let $Z$ be an $n$-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 $Z$ are a drift vector $\theta$ and covariance matrix $\Sigma$ that characterize movement away from the boundary, plus an $n \times n$ matrix R whose columns specify ``directions of reflection'' - more aptly called ``directions of displacement'' - from the orthant's $n$ boundary faces. The matrix $R$ is assumed to be completely-$S$, which is necessary and sufficient for $Z$ to be a semimartingale. This is the case of primary interest in applications.

For dimension $n = 2$, a simple condition is known to be necessary and sufficient for positive recurrence of $Z$; it involves $R$ and $\theta$ but not $\Sigma$. 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 $n = 3$, 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 $n = 3$. 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
Time: 5:00 P.M.
Location: WeH 6423