scholarly journals Stochastic Current of Bifractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jingjun Guo
Keyword(s):  
Brownian Motion ◽  
Malliavin Calculus ◽  
Skorohod Integral ◽  
Bifractional Brownian Motion

We study the regularity of stochastic current defined as Skorohod integral with respect to bifractional Brownian motion through Malliavin calculus. Moreover, we similarly derive some results in the case of multidimensional multiparameter. Finally, we consider stochastic current of bifractional Brownian motion as a distribution in Watanabe spaces.

scholarly journals MULTIDIMENSIONAL BIFRACTIONAL BROWNIAN MOTION: ITÔ AND TANAKA FORMULAS

2007 ◽  
Vol 07 (03) ◽  
pp. 365-388 ◽  
Author(s):  
KHALIFA ES-SEBAIY ◽  
CIPRIAN A. TUDOR
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Malliavin Calculus ◽  
Stochastic Integrals ◽  
Bifractional Brownian Motion ◽  
Multiple Stochastic Integrals

Using the Malliavin calculus with respect to Gaussian processes and the multiple stochastic integrals, we derive Itô's and Tanaka's formulas for the d-dimensional bifractional Brownian motion.

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.

scholarly journals Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Yuquan Cang ◽  
Junfeng Liu ◽  
Yan Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nonparametric Regression ◽  
Local Time ◽  
Kernel Function ◽  
Malliavin Calculus ◽  
Bandwidth Parameter ◽  
Subfractional Brownian Motion ◽  
Normal Law

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

The sub-bifractional Brownian motion

2013 ◽  
Vol 50 (1) ◽  
pp. 67-121 ◽  
Author(s):  
Charles El-Nouty ◽  
Jean-Lin Journé
Keyword(s):  
Brownian Motion ◽  
Integral Test ◽  
Bifractional Brownian Motion

The sub-bifractional Brownian motion, which is a quasi-helix in the sense of Kahane, is presented. The upper classes of some of its increments are characterized by an integral test.

scholarly journals Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion

2020 ◽  
Vol 12 (1) ◽  
pp. 128-145
Author(s):  
Abdelmalik Keddi ◽  
Fethi Madani ◽  
Amina Angelika Bouchentouf
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonparametric Estimation ◽  
Unknown Function ◽  
Asymptotic Properties ◽  
Kernel Estimator ◽  
Trend Function ◽  
Bifractional Brownian Motion ◽  
Numerical Example

AbstractThe main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}}where {{\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}}} is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.

scholarly journals Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Xichao Sun ◽  
Rui Guo ◽  
Ming Li
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Characterization Theorem ◽  
Bessel Process ◽  
Itô Formula ◽  
Bessel Processes ◽  
Ito Formula ◽  
Bifractional Brownian Motion

Let B = B t 1 , … , B t d t ≥ 0 be a d -dimensional bifractional Brownian motion and R t = B t 1 2 + ⋯ + B t d 2 be the bifractional Bessel process with the index 2 HK ≥ 1 . The Itô formula for the bifractional Brownian motion leads to the equation R t = ∑ i = 1 d ∫ 0 t B s i / R s d B s i + HK d − 1 ∫ 0 t s 2 HK − 1 / R s d s . In the Brownian motion case K = 1 and H = 1 / 2 , X t ≔ ∑ i = 1 d ∫ 0 t B s i / R s d B s i ,   d ≥ 1 is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process X t is not a bifractional Brownian motion unless K = 1 and H = 1 / 2 . We also study some other properties and their application of this stochastic process.

scholarly journals Smoothness and Gaussian Density Estimates for Stochastic Functional Differential Equations with Fractional Noise

2020 ◽  
Vol 8 (4) ◽  
pp. 822-833
Author(s):  
Nguyen Van Tan
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Malliavin Calculus ◽  
Functional Differential Equations ◽  
Stochastic Functional Differential Equations ◽  
Density Estimates ◽  
Fractional Noise ◽  
Gaussian Density ◽  
Functional Differential ◽  
Stochastic Functional

In this paper, we study the density of the solution to a class of stochastic functional differential equations driven by fractional Brownian motion. Based on the techniques of Malliavin calculus, we prove the smoothness and establish upper and lower Gaussian estimates for the density.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

2021 ◽  
Vol 24 (02) ◽  
pp. 2150010
Author(s):  
Jingqi Han ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Fractional Brownian Motion ◽  
Stochastic Heat Equation ◽  
Index Formula ◽  
Hurst Index ◽  
Skorohod Integral ◽  
Fractional Brownian Motions ◽  
Infinite Dimensional ◽  
Lp Theory

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

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