Ergodicity of scalar stochastic differential equations with Hoelder continuous coeffcients

Hoang Duc Luu, Tat Dat Tran, and Jürgen Jost

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Submission date: 13. Aug. 2016
Pages: 20
published in: Stochastic processes and their applications, 128 (2018) 10, p. 3253-3272 
DOI number (of the published article): 10.1016/j.spa.2017.10.014
Bibtex
Keywords and phrases: stationary distributions, invariant measures, fokker-planck equation, kullback-leibler divergence, Cox-Ingersoll-Ross model, Ait-Sahalia model, Wright-Fisher model
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Abstract:
It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coeffcient functions satisfying the assumptions of the Yamada-Watanabe theorem [30, 31] and the Feller test for explosions [17, 18], 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 R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called 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.