Stochastic calculus with respect to Gaussian processes
type: Item
instance of: scholarly article
title: Stochastic calculus with respect to Gaussian processes
zbMATH Open document ID: 1015.60047
author: Elisa Alòs
author: Olivier Mazet
author: David Nualart
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors consider a family of Gaussian processes \((B_t)_{t\in {\mathbb R}_+}\) of the form \(B_t = \int_0^t K(t,s) dW_s\), where \(K\) is a deterministic kernel and \((W_t)_{t\in {\mathbb R}_+}\) is a standard Wiener process. They construct a stochastic calculus with respect to such processes via the stochastic calculus of variations, using the anticipating Skorokhod integral operator with respect to \((W_t)_{t\in {\mathbb R}_+}\), which is denoted by \(\delta\). The stochastic integral of an adapted process \(u\) with respect to \((B_t)_{t\in {\mathbb R}}\) is defined to be \(\delta (K^* u)\), where \(K^*\) is the adjoint of the operator with kernel \(K\). Itô and Stratonovich change of variable formulas and Hölder regularity results are proved for indefinite integrals with respect to \((B_t)_{t\in {\mathbb R}}\), for a wide class of deterministic (singular and regular) kernels \(K\). The results apply in particular to fractional Brownian motion with Hurst parameter \(H\in (1/4,1/2)\).
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 1906000
zbMATH Keywords: stochastic integrals
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Itô formula
zbMATH Keywords: fractional Brownian motion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1008956692
OpenAlex ID: W2127436039
cites work: An extension of Itô's formula for anticipating processes
cites work: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2
cites work: Fractional Brownian motion and the Markov property
cites work: Long memory continuous time models
cites work: Itô's formula with respect to fractional Brownian motion and its application
cites work: Stochastic analysis of the fractional Brownian motion
cites work: Fractional Brownian motion: theory and applications
cites work: Stochastic Calculus for Fractional Brownian Motion I. Theory
cites work: On fractional Brownian processes
cites work: L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel
cites work: Existence and uniqueness theorems for fBm stochastic differential equations
cites work: Stochastic analysis of fractional brownian motions
cites work: Q4337939
cites work: Fractional Brownian Motions, Fractional Noises and Applications
cites work: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions
cites work: Q4842684
cites work: Q4218974
cites work: Stochastic calculus with anticipating integrands
cites work: Q3141895
cites work: On a Generalization of a Stochastic Integral
cites work: Integration with respect to fractal functions and stochastic calculus. I
cites work: Integration with respect to fractal functions and stochastic calculus. II
DOI: 10.1214/AOP/1008956692
description: scientific article; zbMATH DE number 1906000