Large Deviations of the Exit Measure through a Characteristic Boundary for a Poisson driven SDE

TL;DR

This paper investigates the large deviations of the exit measure for a Poisson-driven stochastic differential equation, extending methods from Brownian SDEs to Poisson processes and utilizing recent large deviation results.

Contribution

It adapts the large deviation approach for Brownian SDEs to Poisson-driven SDEs, providing new insights into exit measures through characteristic boundaries.

Findings

Extended large deviation techniques to Poisson SDEs

Analyzed exit measure behavior at the boundary of attraction basin

Connected Poisson SDE exit analysis with previous Brownian SDE methods

Abstract

Let O the basin of attraction of the unique stable equilibrium of a dynamical system, which is the law of large numbers limit of a Poissonian SDE. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of M. Day (1990) for the same problem for a Brownian SDE. For that purpose, we will use the Large deviation for the Poissonian SDE reflected at the boundary of O studied in our recent work Pardoux and Samegni (2018).