Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise
type: Item
instance of: scholarly article
title: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise
zbMATH Open document ID: 1076.60054
author: B. Maslowski
author: T. E. Duncan
author: B. Pasik-Duncan
published in: Stochastic Processes and their Applications
publication date: 2005-09-02T00:00:00Z
review text: The article is devoted to the stochastic differential equation \[ dX(t)=A(t)X(t)dt+\sum _{k=1}^mB_kX(t)d\beta ^H_k(t).\tag{1} \] Here \(A(t), B_k, k=1,\dots,m\), are unbounded densely defined linear operators in the infinite-dimensional Hilbert space. The processes \(\beta ^H_k, k=1,\dots,m,\) are independent one-dimensional fractional Brownian motions with the Hurst parameter \(H.\) The authors propose the representation for the strong solution of (1). They suppose, in particular, that the operators \(B_k, k=1,\dots,m,\) generate mutually commuting strongly continuous groups \(S_k(s), k=1,\dots,m,\) and the family \(\tilde{A}(t)=A(t)-Ht^{2H-1}\sum _{k=1}^mB_k^2\) generates a strongly continuous evolution family of operators \(\{U(t,s); 0\leq s\leq t\leq T\}.\) Then the solution of (1) can be obtained by the following formula \[ X(t)=\prod _{k=1}^mS_k(\beta ^H_k(t))U(t,0)x_0. \] Such representation allows to study the stability of the solution using the law of the iterated logarithm for fractional Brownian motion and some kind of Lyapunov function despite of the absence of the Markov property of \(X.\) Examples, when \(\beta ^H\) can have the stabilizing influence, are also presented.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 2201794
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: stability
zbMATH Keywords: stochastic differential equations in Hilbert space
reviewed by: A. A. Dorogovtsev
MaRDI profile type: Publication
full work available at URL: j.spa.2005.03.011
OpenAlex ID: W2039655644
cites work: Stochastic integration with respect to the fractional Brownian motion
cites work: Stochastic Equations in Infinite Dimensions
cites work: Extending a theorem of A. M. Liapunov to Hilbert space
cites work: Some applications of fractional Brownian motion to linear systems
cites work: Stochastic Calculus for Fractional Brownian Motion I. Theory
cites work: STOCHASTIC INTEGRATION FOR FRACTIONAL BROWNIAN MOTION IN A HILBERT SPACE
cites work: FRACTIONAL BROWNIAN MOTION AND STOCHASTIC EQUATIONS IN HILBERT SPACES
cites work: A parabolic stochastic differential equation with fractional Brownian motion input
cites work: Heat equations with fractional white noise potentials
cites work: Stochastic partial differential equations driven by multiparameter fractional white noise
cites work: General Fractional Multiparameter White Noise Theory and Stochastic Partial Differential Equations
cites work: Random Fourier Transforms
cites work: Stability of semilinear stochastic evolution equations
cites work: Fractional Brownian Motions, Fractional Noises and Applications
cites work: Evolution equations driven by a fractional Brownian motion
cites work: Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Stochastic evolution equations with a spatially homogeneous Wiener process
cites work: Q3207428
cites work: Stochastic evolution equations with fractional Brownian motion
cites work: Integration with respect to fractal functions and stochastic calculus. I
DOI: 10.1016/J.SPA.2005.03.011
description: scientific article