Discretization of the Wiener-process in difference-methods for stochastic differential equations
type: Item
instance of: scholarly article
title: Discretization of the Wiener-process in difference-methods for stochastic differential equations
zbMATH Open document ID: 0548.60063
DOI: 10.1016/0304-4149(84)90306-5
author: N. E. Zubov
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: The solution of the stochastic differential equation \(dx(t)=a(t, x(t))dt+b(t,x(t))dw(t)\), \(x(0)=x_ 0\), can be approximated by the Euler- method \(x_{n+1}=x_ n+a(t_ n,x_ n)\Delta t+b(t_ n,x_ n)\Delta w(t_ n)\), if the coefficients are uniformly Lipschitz-continuous. In numerical computations an additional approximation is necessary: the Wiener-process has to be replaced by a suitable simulation. In this paper the effect of this simulation is analysed and the error estimated in terms of the bounded-Lipschitz-distance for measures on \(R^ n\). For diffusion coefficients b, independent of x, the analysis is quite simple but problems arise if this is not the case due to multiplicative effects on the error for each time step. It would be of interest to estimate the distance between laws of the diffusion process and its computer-approximation on the space of continuous functions C[0,T] for a 'suitable' metric on the space of measures on C[0,T]. A 'suitable' metric has not yet been found but this problem could be a direction for further research.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 65C20
Mathematics Subject Classification ID: 65C99
zbMATH DE Number: 3874352
zbMATH Keywords: Euler-method
zbMATH Keywords: bounded-Lipschitz-distance
zbMATH Keywords: computer-approximation
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90306-5
OpenAlex ID: W2000502750
cites work: Q4189915
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description: scientific article; zbMATH DE number 3874352