Precise asymptotics for large deviations of integral forms

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.