The local linearization scheme for nonlinear diffusion models with discontinuous coefficients
type: Item
instance of: scholarly article
title: The local linearization scheme for nonlinear diffusion models with discontinuous coefficients
zbMATH Open document ID: 0979.60055
DOI: 10.1016/S0167-7152(98)00191-6
author: O. Stramer
published in: Statistics \& Probability Letters
publication date: 2000-07-02T00:00:00Z
review text: Consider the stochastic differential equation in \(\mathbb R^d\) \[ dX(t)=b(X(t)) dt+\sigma(X(t)) dB(t), \] where \(B\) is a standard Brownian motion. The author studies a numerical scheme called local linearisation to approximate the solution, which is different from the usual Itô-Taylor numerical schemes. On a time interval \([kh,(k+1)h]\), the diffusion coefficient is assumed to be constant, and the drift is linearised as follows: From Itô formula, \(db(X(t))=J(X(t)) d(X(t))+M(X(t)) dt\), for some \(J\) and \(M\), one can approximate \[ b(X(t))\approx J(X(kh))X(t)+M(X(kh))t+b(X(kh))-J(X(kh))X(kh)-M(X(kh))kh , \] thus obtaining a linear s.d.e. that can be explicitly solved in each interval. The paper gives two conditions for the weak convergence of the approximated diffusion to the solution of the original equation. The coefficients need not be continuous, though it must be assumed that a solution exists and is unique. Essentially, the drift can be piecewise twice-differentiable, and the diffusion coefficient can be only piecewise continuous. Nothing is said about the order of convergence, but some experiments are shown where this scheme behaves better than the Euler scheme. The experiments are performed on continuous-time threshold autoregressive models.
reviewed by: Aureli Alabert
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 65C30
zbMATH DE Number: 1395128
zbMATH Keywords: weak approximations
zbMATH Keywords: numerical schemes
zbMATH Keywords: local linearisation
zbMATH Keywords: continuous-time threshold autoregressive model
MaRDI profile type: Publication
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full work available at URL: s0167-7152(98)00191-6
OpenAlex ID: W1989715977
description: scientific article; zbMATH DE number 1395128