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The problem of the Taylor-Ito and Taylor-Stratonovich expansions of the Itostochastic processes in a neighborhood of a fixed moment of time is considered.The classical forms of the Taylor-Ito and Taylor-Stratonovich expansions aretransformed to the four new representations, which includes the minimal sets ofdifferent types of iterated Ito and Stratonovich stochastic integrals.Therefore, these representations (the so-called unified Taylor-Ito andTaylor-Stratonovich expansions) are more convenient for constructing ofhigh-order strong numerical methods for Ito stochastic differential equations.Explicit one-step strong numerical schemes with the orders of convergence 1.0,1.5, 2.0, 2.5, and 3.0 based on the unified Taylor-Ito and Taylor-Stratonovichexpansions are derived. Effective mean-square approximations of iterated Itoand Stratonovich stochastic integrals from these numerical schemes areconstructed on the base of the multiple Fourier-Legendre series withmultiplicities 1 to 6.

Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and  its Application to the High-Order Strong Numerical Methods for Ito Stochastic  Differential Equations