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

Abstract

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.