scholarly journals Regularity properties of the stochastic flow of a skew fractional Brownian motion

2020 ◽  
Vol 23 (01) ◽  
pp. 2050005
Author(s):  
Oussama Amine ◽  
David R. Baños ◽  
Frank Proske
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Solution Process ◽  
Variational Calculus ◽  
Stochastic Flow ◽  
Intrinsic Properties ◽  
Regularity Properties ◽  
Fractional Brownian Noise ◽  
Unknown Solution ◽  
Variation Part

In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

scholarly journals Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

2021 ◽  
Author(s):  
David Baños ◽  
Salvador Ortiz-Latorre ◽  
Andrey Pilipenko ◽  
Frank Proske
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Solution Process ◽  
Variational Calculus ◽  
Strong Solutions ◽  
Skew Brownian Motion ◽  
Unknown Solution

AbstractIn this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters $$H<\frac{1}{2}.$$ H < 1 2 . Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

A FILTERING PROBLEM FOR A LINEAR STOCHASTIC EVOLUTION EQUATION DRIVEN BY A FRACTIONAL BROWNIAN MOTION

2008 ◽  
Vol 08 (03) ◽  
pp. 397-412
Author(s):  
WILFRIED GRECKSCH ◽  
CONSTANTIN TUDOR
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
Fractional Brownian Motion ◽  
Solution Process ◽  
Innovation Process ◽  
Optimal Estimation ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Finite Dimensional ◽  
Observation Process

A linear unbiased and square mean optimal estimation is obtained for the mild solution process of a stochastic evolution equation with an infinite-dimensional fractional Brownian motion as noise and the noise in the observation process is a finite-dimensional Brownian motion. An innovation process is introduced and the estimation is obtained as a solution of a stochastic differential equation with a finite-dimensional noise. By using an approach based on the equivalence with a deterministic control problem, the estimation for the Fourier coefficients of the signal process is also determined.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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