Runge-Kutta methods for stochastic differential equations

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We have considered numerical methods for ordinary differential equations with noise terms. In general, such methods need a large number of (pseudo-) random numbers and a large number of evaluations of the diffusion coefficients in the equations. In this research subject, numerical schemes with high precision have been proposed in which the sum of the both numbers is smaller than existing schemes need. Incidentally, because numerical calculations cause numerical errors, methods less influenced by the errors are desirable. Such methods with high precision have been also proposed.