The law of the Euler scheme for stochastic differential equations. I: Convergence rate of the distribution function
type: Item
instance of: scholarly article
title: The law of the Euler scheme for stochastic differential equations. I: Convergence rate of the distribution function
zbMATH Open document ID: 0838.60051
DOI: 10.1007/BF01303802
author: Vlad Bally
author: Denis Talay
publication date: 1996-05-27T00:00:00Z
review text: We study the approximation problem of \(\mathbb{E} f(X_T)\) by \(\mathbb{E} f(X^n_T)\), where \((X_t)\) is the solution of a stochastic differential equation, \((X^n_t)\) is defined by the Euler discretization scheme with step \(T/n\), and \(f\) is a given function. For smooth \(f\)'s, Talay and Tubaro have shown that the error \(\mathbb{E} f(X_T) - f(X^n_T)\) can be expanded in powers of \(1/n\), which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when \(f\) is only supposed measurable and bounded, under an additional nondegeneracy condition of Hörmander type for the infinitesimal generator of \((X_t)\): to obtain this result, we use the stochastic variation calculus. In the second part of this work, we will consider the density of the law of \(X^n_T\) and compare it to the density of the law of \(X_T\).
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 849088
zbMATH Keywords: Brownian motion
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: Euler discretization scheme
zbMATH Keywords: convergence rate
MaRDI profile type: Publication
cites work: On the connection between the Malliavin covariance matrix and Hörmander's condition
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full work available at URL: https://doi.org/10.1007/bf01303802
OpenAlex ID: W2607399731
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 849088