An approximate method via Taylor series for stochastic functional differential equations

• Published in: Journal of mathematical analysis and applications Vol. 363; no. 1; pp. 128 - 137
• Main Authors: MILOSEVIC, Marija, JOVANOVIC, Miljana, JANKOVIC, Svetlana
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.03.2010; Elsevier
• Subjects: [formula omitted] and almost sure convergence; Acceleration of convergence; Calculus of variations and optimal control; Exact sciences and technology; Fréchet derivative; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Sciences and techniques of general use; Stochastic functional differential equation; Taylor approximation; Stochastic functional differential equation; Taylor approximation; Fréchet derivative; L p and almost sure convergence; Partition; Approximation; Approximate method; Fréchet differentiability; Taylor series; Probability distribution; Stochastic method; L; Convergence; Functional; Mathematical analysis; Analytical method; and almost sure convergence; Convergence rate; Time interval; Approximate solution; Application
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2009.07.061

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the L p -norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.