FRACTIONAL BROWNIAN MOTION AND STOCHASTIC EQUATIONS IN HILBERT SPACES

2002 ◽  
Vol 02 (02) ◽  
pp. 225-250 ◽  
Author(s):  
T. E. DUNCAN ◽  
B. PASIK-DUNCAN ◽  
B. MASLOWSKI
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Stochastic Equations ◽  
Sample Paths ◽  
Hyperbolic Differential Equations

In this paper, stochastic differential equations in a Hilbert space with a standard, cylindrical fractional Brownian motion with the Hurst parameter in the interval (1/2,1) are investigated. Existence and uniqueness of mild solutions, continuity of the sample paths and state space regularity of the solutions, and the existence of limiting measures are verified. The equivalence of the probability laws for the solution evaluated at different times and different initial conditions and the convergence of these probability laws to the limiting probability are verified. These results are applied to specific stochastic parabolic and hyperbolic differential equations. The solution of a specific parabolic equation with the fractional Brownian motion only in the boundary condition is shown to have many results that are analogues of the results for a fractional Brownian motion in the domain.

RANDOM ATTRACTORS FOR STOCHASTIC EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION

2010 ◽  
Vol 20 (09) ◽  
pp. 2761-2782 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
B. MASLOWSKI ◽  
B. SCHMALFUß
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Stochastic Equations ◽  
Hurst Parameter ◽  
Random Dynamical System ◽  
Random Attractors

In this paper, the asymptotic behavior of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H > 1/2 is studied. In particular, it is shown that the corresponding solutions generate a random dynamical system for which the existence and uniqueness of a random attractor is proved.

Stochastic partial functional integrodifferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior

2019 ◽  
Vol 27 (2) ◽  
pp. 107-122
Author(s):  
Fulbert Kuessi Allognissode ◽  
Mamadou Abdoul Diop ◽  
Khalil Ezzinbi ◽  
Carlos Ogouyandjou
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Resolvent Operator ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter

Abstract This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion {S_{Q}^{H}(t)} , with Hurst parameter {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufficient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory.

scholarly journals Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xia Zhou ◽  
Dongpeng Zhou ◽  
Shouming Zhong
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Fractional Brownian Motion ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Banach Fixed Point Theorem ◽  
Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

scholarly journals Fuzzy stochastic differential equations driven by fractional Brownian motion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Marek T. Malinowski ◽  
M. J. Ebadi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Financial Models ◽  
Stochastic Integral ◽  
Long Range Dependence ◽  
World Systems

AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

Deplay BSDEs driven by fractional Brownian motion

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Sadibou Aidara ◽  
Ibrahima Sane
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
The Past ◽  
Type Integral ◽  
Divergence Type ◽  
Uniqueness Of A Solution

Abstract This paper deals with a class of deplay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {\frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout this paper is the divergence-type integral.

scholarly journals Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

2020 ◽  
Vol 18 (1) ◽  
pp. 1097-1112
Author(s):  
Louk-Man Issaka ◽  
Mamadou Abdoul Diop ◽  
Hasna Hmoyed
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Analysis ◽  
Mild Solutions ◽  
Hurst Parameter ◽  
Resolvent Operators ◽  
Final Point ◽  
Analysis Theory ◽  
Non Local

Abstract This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

On mild solutions of gradient systems in Hilbert spaces

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Andrzej Rozkosz
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Representation ◽  
Infinite Dimensional ◽  
The Cauchy Problem

AbstractWe consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.

scholarly journals Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yousef Alnafisah ◽  
Hamdy M. Ahmed
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Neutral Delay ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<p style='text-indent:20px;'>In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.</p>

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