General diffusion processes as limit of time-space Markov chains

Abstract

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 $(1/4)\wedge (1/\mathit{p})$ 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 $(1/2)\wedge (2/\mathit{p})$ if the grid is adapted to the speed measure of the diffusion, which is optimal for $\mathit{p}\le 4$. 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.