A note on Newton’s method for stochastic differential equations and its error estimate

Abstract

Kawabata and Yamada [3] proposed an \textit{implicit} Newton method for stochastic differential equations and proved its convergence. They proved an error estimate in a sufficiently small time interval and extended it to a global convergence theorem by using open-closed method. In this note, the author gives an \textit{explicit} Newton scheme which is equivalent to Kawabata-Yamada’s \textit{implicit} formulation (Remark~1) and prove its direct error estimate (Theorem~2.1). His result could provide a key to solve the open problem of second order convergence (see Remark of Theorem~2.1 and [2]).