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In this short paper, we consider discrete-time Markov chains on lattices asapproximations to continuous-time diffusion processes. The approximations canbe interpreted as finite difference schemes for the generator of the process.We derive conditions on the diffusion coefficients which permit transitionprobabilities to match locally first and second moments. We derive a novelformula which expresses how the matching becomes more difficult for larger(absolute) correlations and strongly anisotropic processes, such thatinstantaneous moves to more distant neighbours on the lattice have to beallowed. Roughly speaking, for non-zero correlations, the distance covered inone timestep is proportional to the ratio of volatilities in the twodirections. We discuss the implications to Markov decision processes and theconvergence analysis of approximations to Hamilton-Jacobi-Bellman equations inthe Barles-Souganidis framework.

The non-locality of Markov chain approximations to two-dimensional diffusions