Asymptotic expansions for Laplace transforms of Markov processes

Let ${\mu }^{?}$ be the probability measures on $D[0,T]$ of suitable Markov processes ${\{{\xi }_t^{?}\}}_{0\le t\le T}$ (possibly with small jumps) depending on a small parameter $?>0$, where $D[0,T]$ denotes the space of all functions on $[0,T]$ which are right continuous with left limits. In this paper we investigate asymptotic expansions for the Laplace transforms ${\int }_{D[0,T]}\exp ?\{{?}^{?1}F(x)\}{\mu }^{?}(dx)$ as $?\to 0$ for smooth functionals F on $D[0,T]$. This study not only recovers several well-known results, but more importantly provides new expansions for jump Markov processes. Besides several standard tools such as exponential change of measures and Taylor's expansions, the novelty of the proof is to implement the expectation asymptotic expansions on normal deviations which were recently derived in [13].