Sampling from Unknown Transition Densities of Diffusion processes

TL;DR

This paper presents a novel sampling method for diffusion process transition densities, including those without closed-form solutions, by solving a related PDE, with applications to Wright-Fisher models in genetics.

Contribution

The paper introduces a PDE-based sampling technique for unknown transition densities of diffusion processes, extending applicability to complex models like Wright-Fisher diffusions.

Findings

Method performs well on processes with known densities, matching theoretical results.

Successfully applied to Wright-Fisher diffusions relevant in population genetics.

Provides a new tool for sampling in stochastic differential equations with unknown densities.

Abstract

In this paper, we introduce a new method of sampling from transition densities of diffusion processes including those unknown in closed forms by solving a partial differential equation satisfied by the quotient of transition densities. We demonstrate the performance of the developed method on processes with known densities and the obtained results are consistent with theoretical values. The method is applied to Wright-Fisher diffusions owing to their importance in population genetics in studying interaction networks inherent in genetic data. Diffusion processes with bounded drift and non degenerate diffusion are considered as reference processes. $\bf {Key words}:$ Stochastic differential equation (SDE), Transition density, Fokker-Planck partial differential equation, Aronson's bound, Rejection sampling, Wright-Fisher diffusion.