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.

On limit theorems of some extensions of fractional Brownian motion and their additive functionals

2017 ◽  
Vol 17 (03) ◽  
pp. 1750022
Author(s):  
M. Ait Ouahra ◽  
S. Moussaten ◽  
A. Sghir
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Limit Theorems ◽  
Fractional Derivatives ◽  
Slowly Varying Function ◽  
Bifractional Brownian Motion ◽  
Additive Functionals ◽  
Subfractional Brownian Motion

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

scholarly journals Renormalization Of The Local Time For The d-Dimensional Fractional Brownian Motion With N Parameters

2007 ◽  
Vol 186 ◽  
pp. 173-191 ◽  
Author(s):  
M. Eddahbi ◽  
R. Lacayo ◽  
J. L. Solé ◽  
J. Vives ◽  
C. A. Tudor
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Space Variable ◽  
Fixed Time ◽  
Sobolev Norm

AbstractWe study the asymptotic behavior in Sobolev norm of the local time of the d-dimensional fractional Brownian motion with N-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.

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 A DYNAMICAL APPROACH TO FRACTIONAL BROWNIAN MOTION

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 81-94 ◽  
Author(s):  
RICCARDO MANNELLA ◽  
PAOLO GRIGOLINI ◽  
BRUCE J. WEST
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Correlation Function ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
The Other ◽  
Hurst Coefficient ◽  
And Diffusion ◽  
Clear Relation

Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for "Gaussian" processes, our conclusions may well apply to a wider class of systems. On the other hand, systems exist for which scaling might not hold, so we speculate on the possible consequences of the various relations derived in the paper on such systems.

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (02) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

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