  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.
