On the convergence of the Lie-Trotter formula for stochastic differential equations
type: Item
instance of: scholarly article
title: On the convergence of the Lie-Trotter formula for stochastic differential equations
zbMATH Open document ID: 0854.60054
published in: Annales Mathématiques Blaise Pascal
publication date: 1997-01-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79106
review text: The authors consider the one-dimensional stochastic differential equation \[ dX_t = \sigma (t,X_t) dB_t - b(t,X_t) dt, \quad X_0 = x, \] on \([0,T]\) together with the corresponding Lie-Trotter formula that defines approximate solutions \(X^n_t\) and \(Y_t^n\) for \(t\) belonging to an interval \([ih, ih + h]\) of an \(n\)-point equidistant grid on \([0,T]\) by \[ X^n_t = Y^n_{ih} - \int^t_{ih} b(s, X^n_s) ds, \quad Y^n_t = X^n_t + \int^t_{ih} \sigma (s, Y_s^n) dB_s. \] It is proved that if \(b\) and \(\sigma\) are continuous and bounded and a pathwise unique solution \(X\) exists, then both \(X^n\) and \(Y^n\) converge towards \(X\) in the norm of the \(L_2\)-space of \(C[0,T]\)-valued random variables, uniformly for all initial values \(x\) in a compact.
reviewed by: M.Gelbrich
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 65C99
zbMATH DE Number: 784552
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: approximation
zbMATH Keywords: convergence of approximate solutions
zbMATH Keywords: Lie-Trotter formula
author: M. Erraoui
author: Youssef Ouknine
MaRDI profile type: Publication
OpenAlex ID: W2322967611
cites work: Approximation of some stochastic differential equations by the splitting up method
cites work: Q3959169
cites work: Q3792001
cites work: Q3791995
cites work: Q5524051
DOI: 10.5802/AMBP.9
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description: scientific article; zbMATH DE number 784552