Precise asymptotics for large deviations of integral forms

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.