Large Deviations for Additive Functionals of Reflected Jump-Diffusions

TL;DR

This paper analyzes the long-term behavior and rare event probabilities of reflected jump-diffusions in bounded domains, providing a characterization of large deviations via a PIDE and demonstrating numerical solutions on examples.

Contribution

It introduces a novel large deviation framework for reflected jump-diffusions, including a PIDE-based characterization and practical numerical methods.

Findings

Derived a large deviation rate function for reflected jump-diffusions.

Developed a numerical approach to solve the associated PIDE.

Validated methods on examples like Brownian motion and biochemical models.

Abstract

We consider a jump-diffusion process on a bounded domain with reflection at the boundary, and establish long-term results for a general additive process of its path. This includes the long-term behaviour of its occupation time in the interior and on the boundaries. We derive a characterization of the large deviation rate function which quantifies the rate of exponential decay of probabilities of rare events for paths of the process. The characterization relies on a solution of a partial integro-differential equation (PIDE) with boundary constraints. We develop a practical implementation of our results in terms of a numerical solution for the PIDE. We illustrate the method on a few standard examples (Brownian motion, birth-death processes) and on a particular jump-diffusion model arising from applications to biochemical reactions.