Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

Luu Hoang Duc, Tat Dat Tran and Jürgen Jost

Stochastic Processes and their Applications, 2018, vol. 128, issue 10, 3253-3272

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 (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), 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–Émery 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.

Keywords: Stationary distributions; Invariant measure; Fokker–Planck equation; Kullback–Leibler divergence; Cox–Ingersoll–Ross model; Ait-Sahalia model; Wright–Fisher model (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1016/j.spa.2017.10.014

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