Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Abstract

Consider a non-symmetric generalized diffusion X(⋅) in ℝ^d determined by the differential operator $A(\mbox{\boldmath{$x$}})=-\sum_{ij}\partial_{i}a_{ij}(\mbox{\boldmath{$x$}})\partial_{j} +\sum_{i} b_{i}(\mbox{\boldmath{$x$}})\partial_{i}$. In this paper the diffusion process is approximated by Markov jump processes X _n (⋅), in homogeneous and isotropic grids G _n ⊂ℝ^d, which converge in distribution in the Skorokhod space D([0,∞),ℝ^d) to the diffusion X(⋅). The generators of X _n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d≥3 can be applied to processes for which the diffusion tensor $\{a_{ij}(\mbox{\boldmath{$x$}})\}_{11}^{dd}$ 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 (⋅). For piece-wise constant functions a _ij on ℝ^d and piece-wise continuous functions a _ij on ℝ² the construction and principal algorithm are described enabling an easy implementation into a computer code.