Strong Approximations for Markov Chains Weakly Converging to Diffusions

In this paper, we construct strong approximations for discrete-time Markov chains weakly converging to continuous diffusion processes, as well as for their perturbed counterparts. Under the assumption of bounded coefficients, we construct closely coupled versions of these processes on a shared probability space. In particular, for both non-degenerate and degenerate cases, we maximize the probability of their exact pathwise coincidence on discrete time grids. Moreover, we construct such probability space that the probability of a large deviation of the interpolated Markov chain from the continuous diffusion trajectory is small on the entire time interval.