Kawabata and Yamada  proposed an implicit Newton's method for nonlinear stochastic differential equations and proved its convergence. Later Amano  gave an explicit formulation of method and showed its direct error estimate. In this article, we prove a probabilistic second-order error estimate which has been an open problem since 1991.
Submitted June 27, 2011. Published January 4, 2012.
Math Subject Classifications: 60H10, 65C30.
Key Words: Newton's method; stochastic differential equation; second order error estimate.
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| Kazuo Amano |
Department of Mathematics
Faculty of Engineering, Gunma University
Kiryu, 376-8515, Japan
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