Diffusion approximations of some stochastic difference equations revisited
type: Item
instance of: scholarly article
title: Diffusion approximations of some stochastic difference equations revisited
zbMATH Open document ID: 0654.60060
DOI: 10.1016/0304-4149(88)90034-8
author: Hisao Watanabe
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(\{X_ n^{\epsilon}\}\) be stochastic processes with discrete parameters which are asymptotically expressed in the form \[ X^{\epsilon}_{n+1}-X_ n^{\epsilon}=\epsilon F(X_ n,n+1,\omega)+o(\epsilon^ 2)\quad (n=0,1,2,...),\quad X_ 0^{\epsilon}=X_ 0(\omega)\in R^ d, \] where F(x,n,\(\omega)\) \((E(F(x,n,\omega))=0)\) are certain random fields on a probability space. Then the stochastic process generated by linear interpolation of \(X_ n^{\epsilon}\) converges weakly to a diffusion process. The results are known more or less by \textit{H. J. Kushner} and \textit{Hai Huang} [SIAM J. Anal. Math. 40, 528-541 (1981; Zbl 0458.60074)] and the author [Hiroshima Math. J. 14, 15-34 (1984; Zbl 0571.60074)]. In this paper, the author introduces a kind of potential function which is similar to one introduced by Gordin and improves on the previous results.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 4067995
zbMATH Keywords: diffusion approximations
zbMATH Keywords: stochastic difference equations
zbMATH Keywords: martingale methods
zbMATH Keywords: perturbation methods
zbMATH Keywords: potential function
reviewed by: Humihiko Watanabe
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90034-8
OpenAlex ID: W2083898272
cites work: Q3721531
cites work: A limit theorem for turbulent diffusion
cites work: Lectures on stochastic flows and applications. Delivered at the Indian Institute of Science, Bangalore, under the T.I.F.R.-I.I.Sc. programme in applications of mathematics. Notes by M. K. Ghosh
cites work: On the Weak Convergence of a Sequence of General Stochastic Difference Equations to a Diffusion
cites work: Almost sure invariance principles for partial sums of weakly dependent random variables
cites work: Q3862204
cites work: Diffusion approximations of some stochastic differential equations. II
description: scientific article; zbMATH DE number 4067995