Precise asymptotics for large deviations of integral forms

Abstract

For suitable families of locally infinitely divisible Markov processes $\{{\xi }_t^{\epsilon }{\}}_{0\le t\le T}$ 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.

43 pages