Numerical Solutions of Neutral Stochastic Functional Differential Equations

This paper examines the numerical solutions of neutral stochastic functional differential equations ( NSFDEs) $d[x(t)\, - \,u(x_t )]\, = \,f(x_t )dt\, + \,g(x_t )dw(t),\,t \ge \,0$. The key contribution is to establish the strong mean square convergence theory of the Euler- Maruyama approximate solu...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 46; no. 4; pp. 1821 - 1841
Main Authors: Wu, Fuke, Mao, Xuerong
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2008
Subjects:
Approximation
Brownian motion
Determinism
Differential equations
Integers
Lipschitz condition
Mathematical theorems
Numerical analysis
Perceptron convergence procedure
Random variables
Uniqueness
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:This paper examines the numerical solutions of neutral stochastic functional differential equations ( NSFDEs) $d[x(t)\, - \,u(x_t )]\, = \,f(x_t )dt\, + \,g(x_t )dw(t),\,t \ge \,0$. The key contribution is to establish the strong mean square convergence theory of the Euler- Maruyama approximate solution under the local Lipschitz condition, the linear growth condition, and contractive mapping. These conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here are obtained under quite general conditions. Although the way of analysis borrows from [X. Mao, LMS J. Comput. Math., 6 (2003), pp. 141-161], to cope with $u(x_t )$, several new techniques have been developed.
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ISSN:0036-1429
1095-7170
DOI:10.1137/070697021