Markov Jump Processes Approximating a Nonsymmetric Generalized Diffusion: numerics explained to probabilists

Consider a non-symmetric generalized diffusion \(X(\cdot)\) in \({\bbR}^d\) determined by the differential operator \(A(\msx)=-\sum_{ij} \partial_ia_{ij}(\msx)\partial_j +\sum_i b_i(\msx)\partial_i\). In this paper the diffusion process is approximated by Markov jump processes \(X_n(\cdot)\), in homogeneous and isotropic grids \(G_n \subset {\bbR}^d\), which converge in distribution to the diffusion \(X(\cdot)\). The generators of \(X_n(\cdot)\) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for \(d\geq3\) can be applied to processes for which the diffusion tensor \(\{a_{ij}(\msx)\}_{11}^{dd}\) fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes \(X_n(\cdot)\). For \(d=2\) the construction can be easily implemented into a computer code.