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.