Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Consider a non-symmetric generalized diffusion X(⋅) in ℝd determined by the differential operator A(x) = -Σij ∂iaij(x)∂j + Σi bi(x)∂i. 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 {aij(x)}11dd 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.