Optimal Diffusion Processes

TL;DR

This paper investigates diffusion processes with a focus on optimizing convergence rates given a stationary distribution and variance constraints, revealing that linear drifts and specific Pearson processes are optimal.

Contribution

It introduces a framework for identifying optimal diffusion processes with respect to convergence rate, highlighting linear drifts and Pearson processes of Hypergeometric type as optimal solutions.

Findings

Optimal drift function is linear.

Convergence rate bounded by variance ratio.

Pearson diffusion processes of Hypergeometric type are optimal.

Abstract

Of stochastic differential equations, diffusion processes have been adopted in numerous applications, as more relevant and flexible models. This paper studies diffusion processes in a different setting, where for a given stationary distribution and average variance, it seeks the diffusion process with optimal convergence rate. It is shown that the optimal drift function is a linear function and the convergence rate of the stochastic process is bounded by the ratio of the average variance to the variance of the stationary distribution. Furthermore, the concavity of the optimal relaxation time as a function of the stationary distribution has been proven, and it is shown that all Pearson diffusion processes of the Hypergeometric type with polynomial functions of at most degree two as the variance functions are optimal.