Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

By Martin Hairer, Natesh S. Pillai. 2009

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We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hormander's condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not "look into the future." The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

Subfields: Economics and Econometrics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c2c14eb4cf938648c8724a7a6d3c8797c8fc46e

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Citations in the corpus

Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths (2011, methodology): “(z,w)−Q¯ψ(y,w)=E˜ Z1 0 DψT(Φ[1,T](zs,w+))J s (6.5) 0,·ξds. Here, Js 0,· denotes the linearization of (1.1) with the initial condition zs. If moment bounds for the Jacobian Js 0,t are available, as in [23], we can proceed via a stochastic control argument using a Bismut–Elworthy–Li type formula [13] ”
A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise (2011, background): “Note that the law of Y must coincide with the attracting invariant measure for (1) given in [10], see also [12, 13].”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “Firstly, they are a necessary ingredient if one wants to extend the work of [17] and [18] on the ergodicity of non-Markovian systems.”
Smoothness of the density for solutions to Gaussian rough differential equations (2012, background): “and the interpolation inequality (Lemma A3 of [20]) gives”
Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in $(1/2,1)$ (2017, background): “In the articles [11, 12, 13, 14] the existence of adapted stationary solutions to dissipative finite-dimensional SDEs driven by fBm and their speed of convergence to the stationary state is studied, by extending Markovian notions as strong Feller property, invariant measure and adaptedness to the no”