Time-reversal of reflected Brownian motions in the orthant

TL;DR

This paper characterizes the time-reversal of a broad class of reflected Brownian motions in the orthant, extending previous results and introducing new approximation methods to analyze their dual processes.

Contribution

It provides the first comprehensive description of time-reversal for RBMs beyond the skew-symmetric case, including a novel discrete approximation scheme.

Findings

Time-reversal of RBMs is absolutely continuous with respect to an auxiliary RBM.

Introduces a new discrete approximation scheme for RBMs.

Determines the dual semigroups associated with RBMs.

Abstract

We determine the processes obtained from a large class of reflected Brownian motions (RBMs) in the nonnegative orthant by means of time reversal. The class of RBMs we deal with includes, but is not limited to, RBMs in the so-called Harrison-Reiman class [4] having diagonal covariance matrices. For such RBMs our main result resolves the long-standing open problem of determining the time reversal of RBMs beyond the skew-symmetric case treated by R.J. Williams in [16]. In general, the time-reversed process itself is no longer a RBM, but its distribution is absolutely continuous with respect to a certain auxiliary RBM. In the course of the proof we introduce a novel discrete approximation scheme for the class of RBMs described above, and use it to determine the semigroups dual to the semigroups of such RBMs.