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Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations. (English) Zbl 0724.65135
Path by path numerical approximations are studied for the solutions of stochastic differential equations of Itô’s type and Stratonovich’s type driven by one-dimensional Brownian motion.
Discretization schemes are proposed which, as the usual Runge-Kutta methods, do not involve the derivatives of the coefficients of the equation. These schemes are proved to achieve the maximum order of convergence and one appears to be asymptotically efficient: it minimizes asymptotically the quadratic conditional moment of the error given the partition. The intuitive origin of the schemes is explained and several examples are explicited on whose numerical accuracy and speed with respect to the usual Euler scheme are discussed showing clearly the interest of these methods.

65C99 Probabilistic methods, stochastic differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
60F05 Central limit and other weak theorems
60J65 Brownian motion
93E11 Filtering in stochastic control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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