The composite Milstein methods for the numerical solution of Itô stochastic differential equations
type: Item
instance of: scholarly article
title: The composite Milstein methods for the numerical solution of Itô stochastic differential equations
zbMATH Open document ID: 1221.65018
author: M. A. Omar
author: Abdel-Karim Aboul-Hassan
author: Sherif I. Rabia
published in: Journal of Computational and Applied Mathematics
publication date: 2011-03-09T00:00:00Z
review text: Three composite Milstein methods are proposed for approximating strong solutions of the Itô stochastic differential equation (SDE) \[ dy(t)= f(t, y(t))\,dt+ \sum^d_{j=1} g_j(t, y(t))\,dW_j(t), \] where the \(W_j(t)\) are standard Wiener processes. Starting with the case where \(d=1\), the three methods are constructed by combining a semi-implicit Milstein method and an implicit Milstein method. For a linear test equation, the three methods are shown to converge to the exact solution and then mean-square stability, \(T\)-stability, and \(T(A)\)-stability are examined. Summaries of numerical results for a linear SDE and two nonlinear SDEs are provided which suggest that the proposed methods have better accuracy than a known composite Euler method. Then for the case where \(d=2\) similarly favorable stability properties are reported for a linear test equation and for a Duffing-Van der Pol oscillator driven by white noise.
reviewed by: Melvin D. Lax
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L06
zbMATH DE Number: 5863134
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: composite Milstein method
zbMATH Keywords: composite Euler method
zbMATH Keywords: Milstein methods
zbMATH Keywords: numerical stability
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: j.cam.2010.10.026
OpenAlex ID: W2023329313
cites work: The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations
cites work: Q4004325
cites work: High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations
cites work: Numerical methods for strong solutions of stochastic differential equations: an overview
cites work: Q4860586
cites work: Higher-order implicit strong numerical schemes for stochastic differential equations
cites work: Stochastically stable one-step approximations of solutions of stochastic ordinary differential equations
cites work: Balanced Implicit Methods for Stiff Stochastic Systems
cites work: The composite Euler method for stiff stochastic differential equations
cites work: A note on the balanced method
cites work: Balanced Milstein Methods for Ordinary SDEs
cites work: Split-step backward balanced Milstein methods for stiff stochastic systems
cites work: Implicit Taylor methods for stiff stochastic differential equations
cites work: Stability Analysis of Numerical Schemes for Stochastic Differential Equations
cites work: A note on the stability properties of the Euler methods for solving stochastic differential equations
cites work: Mean-Square and Asymptotic Stability of the Stochastic Theta Method
cites work: Q4000061
DOI: 10.1016/J.CAM.2010.10.026
description: scientific article; zbMATH DE number 5863134