General diffusion processes as the limit of time-space Markov chains

We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to (1/4)(1/p) in terms of the maximum cell size of the grid, for any p-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to (1/2)(2/p) if the grid is adapted to the speed measure of the diffusion, which is optimal for p4. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features and present an application on a local time estimation problem.
Joint work with  A. Lejay and D. Villemonais