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

TL;DR

This paper develops a method to approximate non-symmetric generalized diffusion processes in multiple dimensions using Markov jump processes on regular grids, enabling easier simulation and analysis.

Contribution

It explicitly constructs generators for Markov jump processes that converge to the diffusion, providing a practical simulation approach for non-symmetric diffusions.

Findings

Convergence of Markov jump processes to the diffusion in distribution.

Explicit construction of generators for the jump processes.

Implementation feasibility for $d=2$ in computer simulations.

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.