Additive Fractional Brownian Motion Research Articles

Weak solutions for singular multiplicative SDEs via regularization by noise

We study multiplicative SDEs perturbed by an independent additive fractional Brownian motion. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients.

Distribution dependent SDEs driven by additive fractional Brownian motion

We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter Hin (0,1). We establish strong well-posedness under a variety of assumptions on the drift; these include the choice B(·,μ)=(f∗μ)(·)+g(·),f,g∈B∞,∞α,α>1-12H,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B(\\cdot ,\\mu )=(f*\\mu )(\\cdot ) + g(\\cdot ), \\quad f,\\,g\\in B^\\alpha _{\\infty ,\\infty },\\quad \\alpha >1-\\frac{1}{2H}, \\end{aligned}$$\\end{document}thus extending the results by Catellier and Gubinelli (Stochast Process Appl 126(8):2323–2366, 2016) to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter H>frac {1}{2} and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order mathcal {O}(h^{2}+varDelta t) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

Hurst Index Estimation in Stochastic Differential Equations Driven by Fractional Brownian Motion

We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the quadratic variations of the solution, defined via higher-order increments. Then we apply our results to construct and study estimators for the Hurst index.

Nonparametric inference for fractional diffusion

A non-parametric diffusion model with an additive fractional Brownian motion noise is considered in this work. The drift is a non-parametric function that will be estimated by two methods. On one hand, we propose a locally linear estimator based on the local approximation of the drift by a linear function. On the other hand, a Nadaraya-Watson kernel type estimator is studied. In both cases, some non-asymptotic results are proposed by means of deviation probability bound. The consistency property of the estimators are obtained under a one sided dissipative Lipschitz condition on the drift that insures the ergodic property for the stochastic differential equation. Our estimators are first constructed under continuous observations. The drift function is then estimated with discrete time observations that is of the most importance for practical applications.

Controlling the solution of stochastic differential equations on a plane with additive fractional Brownian motion

Sufficient conditions are proposed for the existence of an optimum strategy for controlling the solution of stochastic differential equations on a plane containing an additive fractional Brownian motion with the Hurst parameter belonging to the interval (0, 1/2)2.

An Infinite-Dimensional Fractional Linear Quadratic Regulator Problem

We consider a controlled linear stochastic infinite-dimensional differential equation with an additive fractional Brownian motion as noise input. An optimal closed-loop control is determined in the case of complete state information and a quadratic goal functional.

On the Control Problem for Solution of Stochastic Differential Equation with Additive Fractional Brownian Motion

Sufficient conditions for existence of optimal strategy of control of solution of the stochastic differential equation, which includes additive fractional Brownian process with the Hurst parameter belonging to the interval (0, 1/2).

Mutual Information for Stochastic Signals and Fractional Brownian Motion

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>The mutual information between a stochastic signal and this signal plus a fractional Brownian motion (described as an additive fractional Gaussian noise channel) is expressed as the error of an estimation problem that can be naturally associated with this model. If the stochastic signal with the additive fractional Brownian motion occurs multiplied by a scalar parameter, then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem. These results generalize some results for a model where the fractional Brownian motion is a Brownian motion to a model with an arbitrary fractional Brownian motion. </para>

A singular stochastic differential equation driven by fractional Brownian motion

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0 .

Ergodicity of stochastic differential equations driven by fractional Brownian motion

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.

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.