Precise asymptotics for large deviations of integral forms

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