Accuracy of Diffusion Approximations for High Frequency Markov Data

Abstract

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Edgeworth type expansions of third order for transition densities are proved. This is done for time horizons that converge to 0. For this purpose we represent the transition density as a functional of densities of sums of i.i.d. variables. This will be done by application of the parametrix method. Then we apply Edgeworth expansions to the densities. The resulting series gives our Edgeworth-type expansion for the transition density of Markov chains. The research is motivated by applications to high frequency data that are available on a very fine grid but are approximated by a diffusion model on a more rough grid.