Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Consider a non-symmetric generalized diffusion X(⋅) in ℝd determined by the differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(\mbox{\boldmath{$x$}})=-\sum_{ij}\partial_{i}a_{ij}(\mbox{\boldmath{$x$}})\partial_{j} +\sum_{i} b_{i}(\mbox{\boldmath{$x$}})\partial_{i}$\end{document}. In this paper the diffusion process is approximated by Markov jump processes Xn(⋅), in homogeneous and isotropic grids Gn⊂ℝd, which converge in distribution in the Skorokhod space D([0,∞),ℝd) to the diffusion X(⋅). The generators of Xn(⋅) 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{a_{ij}(\mbox{\boldmath{$x$}})\}_{11}^{dd}$\end{document} 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 Xn(⋅). For piece-wise constant functions aij on ℝd and piece-wise continuous functions aij on ℝ2 the construction and principal algorithm are described enabling an easy implementation into a computer code.