On some estimates in quasi sure limit theorem for SDE's
type: Item
instance of: scholarly article
title: On some estimates in quasi sure limit theorem for SDE's
zbMATH Open document ID: 0922.60049
DOI: 10.1016/S0007-4497(99)80005-0
published in: Bulletin des Sciences Mathématiques
publication date: 1999-10-14T00:00:00Z
review text: Let \[ dX(t)= \sigma\bigl(X(t)\bigr) dw(t)+ b\bigl(X(t)\bigr)dt, \quad X(0)=X_0,\tag{1} \] be a stochastic differential equation, where \(\sigma:R^d \to R^d\times R^r\) and \(b:R^d\to R^d\) are smooth functions with bounded derivatives of all orders, \(w(t)\) is the \(r\)-dimensional standard Brownian motion. The author considers the Stroock-Varadhan \(w_n(t)\) and Malliavin \(w_\varepsilon (t)\) approximation of Brownian motion. It is shown that for every \(p,r'\) \[ \sup_n\bigl\| X_n(t)- X_n(s)\bigr \|_{p,r'}< C| t-s|^p \quad \text{and} \quad\sup_{\varepsilon}\bigl\| X_\varepsilon(t)-X_\varepsilon (s) \bigr\|_{2p,r'}<C| t-s|^p, \] where \(X_n(t)\) and \(X_\varepsilon(t)\) denote the unique solutions of (1) with \(w_n\) and \(w_\varepsilon\), respectively, and \(\|\cdot\|_{p,r'}\) is the usual Sobolev norm in Malliavin calculus.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1239980
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Stroock-Varadhan approximation
zbMATH Keywords: Malliavin smooth approximation
author: Jiagang Ren
reviewed by: Maria Stolarczyk
MaRDI profile type: Publication
cites work: Q3308798
cites work: Flows of stochastic dynamical systems: The functional analytic approach
cites work: Quasi sure stochastic flows
cites work: Q4197841
cites work: Q3488968
cites work: The Malliavin calculus, a functional analytic approach
cites work: Lectures on stochastic differential equations and Malliavin calculus
full work available at URL: s0007-4497(99)80005-0
OpenAlex ID: W2061137509
description: scientific article; zbMATH DE number 1239980