Ergodicity of scalar stochastic differential equations with H\"older continuous coefficients

TL;DR

This paper proves exponential convergence to stationary distributions for scalar stochastic differential equations with Hölder continuous coefficients, using a geometric approach involving free energy and divergence measures, with applications in finance and genetics.

Contribution

It introduces a novel geometric method to analyze the ergodicity of SDEs with Hölder continuous coefficients, establishing exponential convergence rates.

Findings

Density functions converge exponentially to the stationary density

Relation established between curvature conditions and dissipativity

Applications demonstrated in finance and population genetics

Abstract

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada-Watanabe theorem \cite{yamada1,yamada2} and the Feller test for explosions \cite{feller51,feller54}, there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called {\it free energy function} on the density function space, we prove that the density functions, which are solutions of the Fokker-Planck equation, converge to the stationary density function exponentially under the Kullback-Leibler {divergence}, thus also in the total variation norm. The results show that there is a relation between the Bakry-Emery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher-Lamperti transformation. Several applications are discussed, including the Cox-Ingersoll-Ross model and the Ait-Sahalia model in finance and the Wright-Fisher model in population genetics.