Kawabata and Yamada [3] proposed an 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 explicit Newton scheme which is equivalent to Kawabata-Yamada’s 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]).

In the study of spectral functions of Strum-Liouville operators, S. Kotani found a one-to-one correspondence between the operators and Herglotz functions determining the spectral measures. In the present paper we give another correspondence to a class of Lévy processes obtained as a compensated integrals of Brownian local times. As an application the continuity theorem of Kotani’s correspondence is extended.