O. Yu. Kulchitski
Russia, 195251, St.-Petersburg, Polytechnicheskaja st. 29,
St.-Petersburg
State Technical University,
Department of Mechanics
and Control Processes,
D. F. Kuznetsov
Russia, 195251, St.-Petersburg, Polytechnicheskaja st. 29,
St.-Petersburg
State Technical University,
Department of Mathematics,
For the first time, the Taylor-Ito expansion of Ito processes was obtained in papers by W.Wagner and E.Platen in 1982. The Taylor-Ito expansion includes the special repeated stochastic integrals. These repeated stochastic integrals are chains of stochastic integrals, certain integrals are on the Wiener process in the Ito sense while the others are on the time. In the present paper, we transform the repeated stochastic integrals included in the Taylor-Ito expansion to the repeated stochastic Ito integrals with polynomial functions by changing the order of integration in repeated stochastic integrals. The Taylor-Ito expansion on such stochastic integrals obtained in the present paper is called the unified Taylor-Ito expansion. Certain special interesting recurrence relations between the coefficient functions of the unified Taylor-Ito expansion are stated. These relations show that the unified Taylor-Ito expansion is a natural generalization of the deterministic Taylor expansion for a class of Ito processes. It should be noted that the unified Taylor-Ito expansion include a smaller number of different repeated stochastic integrals than the Taylor-Ito expansion in the form of W.Wagner and E.Platen. Numerical simulation of repeated stochastic integrals is a very difficult theoretical and numerical problem. Therefore, the unified Taylor-Ito expansion is more convenient for numerical solution of stochastic differential Ito equations than the Taylor-Ito expansion suggested by W.Wagner and E.Platen.