Multi-dimensional reflected processes: Boundary characterization and applications to stochastic networks
Abstract
We study multi-dimensional reflected processes, in particular reflected diffusions constrained to lie in [special characters omitted] with signed jumps, mainly motivated from their applications to stochastic networks. We provide a boundary behavior characterization for this class of processes, even in the general setting when the drift and diffusion coefficients, as well as the directions of reflection upon hitting the boundary faces of [special characters omitted], are allowed to be random fields over time and space. We show boundary results known in the case of semi-martingale reflecting Brownian motion (SRBM) continue to hold in this setting. We relate the regulator processes to semi-martingale local times at the boundaries, showing they do not charge the set of times the process expends at the intersection of two or more boundary faces. All these results are established under mild boundary conditions on the diffusion coefficients and under a completely-S structure with an additional invertibility requirement on the reflection matrix. We then use this boundary behavior characterization to provide necessary and sufficient conditions for a product-form distribution in stationary regime, generalizing the known conditions for SRBMs and some negative results in the context of reflected Lévy processes (including jumps). For the case of space-dependent but non-random drift, diffusion and reflection matrix coefficients, and jumps with space-dependent amplitudes driven by Poisson random measures, sufficient conditions for the existence and uniqueness of, as well as convergence to a product-form stationary distribution are provided. A further characterization of such a stationary distribution in terms of semi-martingale local times is accomplished.
Degree
Ph.D.
Advisors
Mazumdar, Purdue University.
Subject Area
Electrical engineering|Mathematics|Statistics
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