The Euler scheme with irregular coefficients
type: Item
instance of: scholarly article
title: The Euler scheme with irregular coefficients
zbMATH Open document ID: 1020.60054
author: Liqing Yan
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The author considers a \(d\)-dimensional system of stochastic differential equations, driven by Brownian motion \(B\) in \(R^r\), \[ X_t = X_0 + \int_0^t b(s,X_s) ds + \int_0^t \sigma(s,X_s) dB_s,\tag{1} \] where \(X_0\) is an \(R^d\)-valued random variable, independent of \(B\). For the coefficient functions he assumes necessary and sufficient conditions such that equation (1) has a unique weak solution, based on a result by \textit{H. J. Engelbert} and \textit{W. Schmidt} [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 69, 143-155 (1985; Zbl 0583.60052)]. He then considers the continuous Euler scheme \[ X_t^n = X_{\tau_k^n}^n + b(\tau_k^n,X_{\tau_k^n}^n)(t-\tau_k^n) + \sigma(\tau_k^n,X_{\tau_k^n}^n)(B_t - B_{\tau_k^n}),\tag{2} \] for \({\tau_k^n} \leq t \tau_{k+1}^n, k=0,1,\ldots,n\), where \(0=\tau_0^n \leq \tau_1^n \leq \ldots \leq \tau_n^n = T\) is a sequence of random partitions of \([0,T]\). The aim of the paper is to show weak convergence of the solution of the continuous Euler scheme (2) to the unique weak solution of (1) under only mild smoothness conditions on the coefficient functions \(b\) and \(\sigma\). Under additional conditions the author proves results concerning the rate of convergence in the one-dimensional case.
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1906082
zbMATH Keywords: Euler scheme
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: weak convergence
zbMATH Keywords: rate of convergence
zbMATH Keywords: Monte Carlo simulations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1029867124
OpenAlex ID: W2026701194
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DOI: 10.1214/AOP/1029867124
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description: scientific article; zbMATH DE number 1906082