Ergodicity of stochastic differential equations driven by fractional Brownian motion

By Martin Hairer. Annals of Probability, 2003

137 citations recorded by the source corpus.

We study the ergodic properties of finite-dimensional syste ms of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter H 2 (0, 1). A general framework is constructed to make precise the notions of “invariant measure” and “stationary state” for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cac03353c90d397696429eb4230d0a587bcbb70a

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Citations in the corpus

A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise (2011, background): “[10] M. Hairer (2005): Ergodicity of stochastic differential equations driven by fractional Brownian motion.”
Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion (2012, background): “In [18] the author investigates the ergodicity of the solution when σ is constant, as well as the convergence rate toward the stationary solution; see also [22] for infinitedimensional evolution equations driven by an fBm in an additive way.”
Ergodicity of the Infinite Dimensional Fractional Brownian Motion (2011, unknown): “tems, given by the so-called stochastic dynamical systems’ framework, see Hairer [ 11 ]a nd Hairer et al. [12].”
Non-degeneracy of Wiener functionals arising from rough differential equations (2007, background): “Such differential equations arise, for instance, in financial mathematics [2, 10, 11] or as model for studying ergodic properties of non-Markovian systems [12].”
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains (2006, background): “One can show by methods from [5] (but see also [29] for a similar problem in the one-dimensional case) that for ξ ∈ (0, 1/2), there exists a Borel probability measure P on Ω(ξ,E) such that the canonical process wt, t ∈ R, defined by (6.”
Parameter estimation for a partially observed Ornstein–Uhlenbeck process with long-memory noise (2015, methodology): “Then to prove Lemma 2, using Birkhoff’s ergodic theorem (for instance see [13]), it is sufficient to study the convergence of E[X2 t ], E[Σ 2 t ] and E[ΣtXt] as”
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”
Densities for rough differential equations under Hormander's condition (2007, methodology): “Differential equations of this type have also been used as simple examples for the study of ergodicity of non-Markovian systemems, [24].”
On the Diffusion of Shape (2007, background): “But, as is noted by the author towards the end of [18], the case of multiplicative noise is considerably more involved than the additive case and requires better estimates.”