ON THE PROPERTIES OF BIFRACTIONAL BROWNIAN MOTION

2021 ◽  
Vol 3 (2) ◽  
pp. 59-67
Author(s):  
BA Demba Bocar
Keyword(s):  
Brownian Motion ◽  
Bifractional Brownian Motion

In this paper, we study several properties of the bifractional Brownian motion introduced by Houdr\'{e} and Villa.

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 Bifractional Brownian motion: existence and border cases

2015 ◽  
Vol 19 ◽  
pp. 766-781 ◽  
Author(s):  
Mikhail Lifshits ◽  
Ksenia Volkova
Keyword(s):  
Brownian Motion ◽  
Bifractional Brownian Motion

scholarly journals On the law of the solution to a stochastic heat equation with fractional noise in time

2015 ◽  
Vol 23 (3) ◽  
Author(s):  
Solesne Bourguin ◽  
Ciprian A. Tudor
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Fractional Brownian Motion ◽  
Gaussian Noise ◽  
Stochastic Heat Equation ◽  
Bifractional Brownian Motion ◽  
Fractional Noise ◽  
The Stochastic Heat Equation ◽  
The Law

AbstractWe study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion. Our result is an extension of a result by Swanson.

Upper classes of the bifractional Brownian motion

2011 ◽  
Vol 48 (3) ◽  
pp. 371-407 ◽  
Author(s):  
Charles El-Nouty ◽  
Jean-Lin Journé
Keyword(s):  
Brownian Motion ◽  
Integral Test ◽  
Bifractional Brownian Motion

Let {BH,K(t); t ≧ 0} be a bifractional Brownian motion with indices 0 < H < 1 and 0 < K ≦ 1. We characterize the upper classes of some increments of BH,K by an integral test.

Sign in / Sign up

Export Citation Format

Share Document