Precise asymptotics for large deviations of integral forms

TL;DR

This paper establishes precise asymptotic formulas for large deviations of integral functionals of locally infinitely divisible Markov processes with small jumps, linking these results to partial integro-differential equations.

Contribution

It provides new asymptotic results for expectations of exponential functionals of Markov processes with small jumps, extending previous asymptotic expansion techniques.

Findings

Derived precise large deviation asymptotics for integral forms

Connected large deviation results to partial integro-differential equations

Utilized recent asymptotic expansion results for expectations

Abstract

For suitable families of locally infinitely divisible Markov processes $\{\xi^{{\epsilon}}_t\}_{0\leq t\leq T}$ with frequent small jumps depending on a small parameter $\epsilon>0,$ precise asymptotics for large deviations of integral forms $\mathbb{E}^{\epsilon}[\exp\{{\epsilon}^{-1}F(\xi^{\epsilon})\}]$ are proved for smooth functionals $F.$ The main ingredient of the proof in this paper is a recent result regarding the asymptotic expansions of the expectations $\mathbb{E}^{\epsilon}[G(\xi^{\epsilon})\}]$ for smooth $G.$ Several connections between these large deviation asymptotics and partial integro-differential equations are included as well.