Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

TL;DR

This paper analyzes large deviations for symmetric reflected Brownian motion in the octant, identifying conditions for optimal paths and simplifying computational approaches for these stochastic processes.

Contribution

It provides new conditions for optimal paths in symmetric SRBM large deviations and demonstrates how to identify spiral versus gradual paths, aiding computational methods.

Findings

Optimal paths can be gradual or spiral around the boundary.

Conditions are derived for when spiral paths are optimal.

Example provided where spiral path is verified as optimal.

Abstract

We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in the positive octant. Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can be verified that a spiral path is optimal. For rotationally symmetric SRBM's, our results facilitate the simplification of computational methods for determining optimal solutions to variational problems and give insight into large deviations behavior of these processes.