Limit theorems with asymptotic expansions for stochastic processes

In this paper, we consider some families of one-dimensional locally infinitely divisible Markov processes {ηtϵ}0≤t≤T with frequent small jumps. For a smooth functional F(x[0,T])F(x[0,T]) on space D[0,T]D[0,T], the following asymptotic expansions for expectations are proved: as ϵ→0ϵ→0,EϵF(ηϵ[0,T])=EF(η0[0,T])+∑i=1sϵi/2EAiF(η0[0,T])+o(ϵs/2) for some Gaussian diffusion η0η0 as the weak limit of ηϵηϵ, suitable differential operators AiAi, and a positive integer ss depending on the smoothness of FF.