On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions

2020 ◽  
pp. 1-32
Author(s):  
Nguyen Huy Tuan ◽  
Tomás Caraballo ◽  
Tran Ngoc Thach
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Wiener Process ◽  
Mild Solution ◽  
Parabolic Equations ◽  
Regularization Method ◽  
Hurst Parameter ◽  
Standard Brownian Motion ◽  
Brownian Motions ◽  
Fractional Brownian Motions

In this paper, we study two terminal value problems (TVPs) for stochastic bi-parabolic equations perturbed by standard Brownian motion and fractional Brownian motion with Hurst parameter h ∈ ( 1 2 , 1 ) separately. For each problem, we provide a representation for the mild solution and find the space where the existence of the solution is guaranteed. Additionally, we show clearly that the solution of each problem is not stable, which leads to the ill-posedness of each problem. Finally, we propose two regularization results for both considered problems by using the filter regularization method.

scholarly journals Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Elhoussain Arhrrabi ◽  
M’hamed Elomari ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Stability Of Solutions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Existence And Stability ◽  
Fractional Stochastic Differential Equations

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.

Moderate deviation for parameter estimator in the stochastic parabolic equations with additive fractional Brownian motion

2014 ◽  
Vol 14 (03) ◽  
pp. 1450002
Author(s):  
Jiang Hui
Keyword(s):  
Brownian Motion ◽  
Evolution Equation ◽  
Fractional Brownian Motion ◽  
Parabolic Equations ◽  
Moderate Deviation ◽  
Hurst Parameter ◽  
Stochastic Evolution Equation ◽  
Stochastic Evolution ◽  
Parameter Estimator ◽  
Asymptotic Behaviors

In this paper, we study the asymptotic behaviors of parameter estimator in a diagonalizable stochastic evolution equation driven by additive fractional Brownian motion with Hurst parameter H ∈ [½, 1). The moderate deviation for this estimator can be obtained.

scholarly journals Transportation Inequalities for Coupled Fractional Stochastic Evolution Equations Driven by Fractional Brownian Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Wentao Zhan ◽  
Yuanyuan Jing ◽  
Liping Xu ◽  
Zhi Li
Keyword(s):  
Fixed Point ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Hurst Parameter ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Uniform Distance

In this paper, we consider the existence and uniqueness of the mild solution for a class of coupled fractional stochastic evolution equations driven by the fractional Brownian motion with the Hurst parameter H∈1/4,1/2. Our approach is based on Perov’s fixed-point theorem. Furthermore, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.

scholarly journals Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zhi Wang ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Wiener Process ◽  
Existence And Uniqueness ◽  
Volterra Equation ◽  
Uniqueness Result ◽  
Hurst Parameter ◽  
Stochastic Volterra Equation

For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameterH>1/2, we prove an existence and uniqueness result for this equation under suitable assumptions.

STOCHASTIC INTEGRATION FOR FRACTIONAL BROWNIAN MOTION IN A HILBERT SPACE

2006 ◽  
Vol 06 (01) ◽  
pp. 53-75 ◽  
Author(s):  
T. E. DUNCAN ◽  
J. JAKUBOWSKI ◽  
B. PASIK-DUNCAN
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Malliavin Calculus ◽  
Stochastic Integration ◽  
Smooth Functions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Stochastic Operator ◽  
Definition Of

A Hilbert space-valued stochastic integration is defined for an integrator that is a cylindrical fractional Brownian motion in a Hilbert space and an operator-valued integrand. Since the integrator is not a semimartingale for the fractional Brownian motions that are considered, a different definition of integration is required. Both deterministic and stochastic operator-valued integrands are used. The approach uses some ideas from Malliavin calculus. In addition to the definition of stochastic integration, an Itô formula is given for smooth functions of some processes that are obtained by the stochastic integration.

OPTIMAL CONSUMPTION AND PORTFOLIO IN A BLACK–SCHOLES MARKET DRIVEN BY FRACTIONAL BROWNIAN MOTION

2003 ◽  
Vol 06 (04) ◽  
pp. 519-536 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL ◽  
AGNÈS SULEM
Keyword(s):  
Mathematical Model ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Consumption Rate ◽  
Hurst Parameter ◽  
Optimal Consumption ◽  
Standard Brownian Motion ◽  
Power Type ◽  
Black Scholes ◽  
Market Driven

We present a mathematical model for a Black–Scholes market driven by fractional Brownian motion BH(t) with Hurst parameter [Formula: see text]. The interpretation of the integrals with respect to BH(t) is in the sense of Itô (Skorohod–Wick), not pathwise (which is known to lead to arbitrage). We find explicitly the optimal consumption rate and the optimal portfolio in such a market for an agent with utility functions of power type. When H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

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.

Malliavin calculus used to derive a stochastic maximum principle for system driven by fractional Brownian and standard Wiener motions with application

2020 ◽  
Vol 28 (4) ◽  
pp. 291-306
Author(s):  
Tayeb Bouaziz ◽  
Adel Chala
Keyword(s):  
Brownian Motion ◽  
Maximum Principle ◽  
Control Problem ◽  
Fractional Brownian Motion ◽  
Stochastic Control ◽  
Malliavin Calculus ◽  
Stochastic Maximum Principle ◽  
Hurst Parameter ◽  
Principle Of Optimality ◽  
Initial Cost

AbstractWe consider a stochastic control problem in the case where the set of the control domain is convex, and the system is governed by fractional Brownian motion with Hurst parameter {H\in(\frac{1}{2},1)} and standard Wiener motion. The criterion to be minimized is in the general form, with initial cost. We derive a stochastic maximum principle of optimality by using two famous approaches. The first one is the Doss–Sussmann transformation and the second one is the Malliavin derivative.

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