Limit theorems with asymptotic expansions for stochastic processes

In this paper, we consider some families of one-dimensional locally infinitely divisible Markov processes { η t ϵ } 0 ≤ t ≤ T with frequent small jumps. For a smooth functional F ( x [ 0 , T ] ) on space D [ 0 , T ] , the following asymptotic expansions for expectations are proved: as ϵ → 0 , E ϵ F ( η ϵ [ 0 , T ] ) = E F ( η 0 [ 0 , T ] ) + ∑ i = 1 s ϵ i / 2 E A i F ( η 0 [ 0 , T ] ) + o ( ϵ s / 2 ) for some Gaussian diffusion η 0 as the weak limit of η ϵ , suitable differential operators A i , and a positive integer s depending on the smoothness of F .