Approximation schemes associated to a differential equation governed by a Hölderian function; the case of fractional Brownian motion.
type: Item
instance of: scholarly article
title: Approximation schemes associated to a differential equation governed by a Hölderian function; the case of fractional Brownian motion.
zbMATH Open document ID: 1075.60073
author: Ivan Nourdin
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-06-13T00:00:00Z
review text: The author considers stochastic differential equations of the form \[ dx_t=\sigma(x_t)dg_t + b(x_t) dt,\;t\in [0,1],\quad x(0)=x_0,\tag{1} \] where \(g:[0,1] \rightarrow R\) is a Hölder function with index \(\alpha \in(0,1]\). He provides a definition of the stochastic integral represented by the term \(\sigma(x_t)dg_t\) in (1) and also of the solution of (1). Two approximation schemes, the Euler and the Milstein method, are proposed and a proof of their convergence is sketched. The case of \(g\) being a trajectory of fractional Brownian motion is discussed as a special case of the previous presentations.
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H35
Mathematics Subject Classification ID: 65C30
zbMATH DE Number: 2175628
zbMATH Keywords: stochastic integral with Hölder function
zbMATH Keywords: Euler method
zbMATH Keywords: Milstein method
describes a project that uses: SimEstFBM
MaRDI profile type: Publication
full work available at URL: j.crma.2005.03.013
OpenAlex ID: W1968010056
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DOI: 10.1016/J.CRMA.2005.03.013
description: scientific article; zbMATH DE number 2175628