Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions
type: Item
instance of: scholarly article
title: Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions
zbMATH Open document ID: 1019.60053
author: Magnus Wiktorsson
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The author considers a system of stochastic differential equations \[ dX(t) = b(X(t),t) dt + \sigma(X(t),t) dW(t), \tag{1} \] where \(X(t)\) is a \(d\)-dimensional vector and \(W(t)\) is an \(m\)-dimensional vector of independent Brownian motions. In the case where the dispersion matrix \(\sigma\) does not satisfy the commutativity condition (which is \(L^i \sigma_{kj}= L^j \sigma_{ki}\), \(i,j = 1,\ldots,m\), \(k=1,\ldots, d\) with \(L^i = \sum_{l=1}^d \sigma_{li} \frac{\partial}{\partial x_l}\)), numerical schemes do not converge in the mean square sense with order larger than \(\frac{1}{2}\). Order \(1\) can be achieved when the iterated stochastic Itô integrals \( I_{i,j}(t_n,t_n+h) = \int_{t_n}^{t_n+h} \int_{t_n}^s dW_i(u) dW_j(s)\) are included (e.g. in the Milstein scheme). An expression for the conditional joint characteristic function of the iterated Itô integrals given the Brownian increments is derived. Moreover, the author suggests a method to approximate \( I_{i,j}\) which can be used in the case \(m>2\) [for \(m=2\) see \textit{J. G. Gaines} and \textit{T. J. Lyons}, SIAM J. Appl. Math. 54, 1132-1146 (1994; Zbl 0805.60052) and \textit{T. Rydén} and the author, Stochastic Processes Appl. 91, 151-168 (2001)], based on the method proposed by \textit{P. E. Kloeden, E. Platen} and \textit{I. W. Wright} [Stochastic Anal. Appl. 10, 431-441 (1992; Zbl 0761.60048)]. The algorithm is essentially a truncation of an infinite series representation of the iterated Itô integral. The mean-square error of the original method is of order \(h^2/n\), where \(n\) is the number of terms in the truncated sum. By a modification of the method, which amounts to adding a term approximating the tail-sum of the discarded terms of the series, the author achieves a mean-square error of order \(h^2/n^2\).
reviewed by: Evelyn Buckwar
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1906244
zbMATH Keywords: iterated Itô integral
zbMATH Keywords: multidimensional stochastic differential equation
zbMATH Keywords: numerical approximation
zbMATH Keywords: variance mixture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345301
OpenAlex ID: W1978024676
cites work: Random Generation of Stochastic Area Integrals
cites work: Q5689624
cites work: Q4714065
cites work: The approximation of multiple stochastic integrals
cites work: Q5807209
cites work: Q5509994
cites work: Q4004320
cites work: Approximate Integration of Stochastic Differential Equations
cites work: On series representations of infinitely divisible random vectors
cites work: Numerical Treatment of Stochastic Differential Equations
cites work: On the simulation of iterated Itô integrals.
cites work: Perks' distributions and their role in the theory of Wiener's stochastic variables
DOI: 10.1214/AOAP/1015345301
Recommended article: Some experiments on numerical simulations of stochastic differential equations and a new algorithm
Recommended article: A second-order Monte Carlo method for the solution of the Ito stochastic differential equation
Recommended article: Iterated stochastic integrals in infinite dimensions: approximation and error estimates
Recommended article: Asymptotically Efficient Runge-Kutta Methods for a Class of Itô and Stratonovich Equations
Recommended article: Simulation of stochastic differential equations
Recommended article: Comparative analysis of high-accuracy numerical methods for solving Itô stochastic differential equations
Recommended article: Q3993249
Recommended article: Q3698003
description: scientific article; zbMATH DE number 1906244