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Русская версия

**D. F. Kuznetsov**

195251, Russia,

Saint Petersburg, Polytechnicheskaya, 29.

Saint Petersburg State Plytechnical university

The monograph proposes the new direction to representations and strong approximations of multiple stochastic Ito and Stratonovich integrals which is based on functional analisys methods. The theorem about expansion of multiple stochastic Ito integrals of multiplicity n (n=1, 2, ... ) based on multiple Fourier series converging in L2([t, T]x ... x[t,T]) (k times) has been proved. This theorem is adapted to multiple stochastic Stratonovich integrals of multiplicity 2 and 3. The theorem about expansion of multiple stochastic Stratonovich integrals of multiplicity n (n=1, 2, ...) based on iterative Fourier series converging pointwise on [t, T] has been proved. Strong approximations of multiple stochastic Ito and Stratonovich integrals for multiplicity 1 - 5 using system of Legendre polynomials and for multiplicity 1 - 3 using system of trigonometric functions have been constructed. Both mean-square convergence and convergence in mean of degree 2n (n=1, 2, ... ) for considering approximations have been proved. Exact formulas for mean-square error of approximation for multiple stochastic Ito integrals of multiplicity 1 - 4 have been obtained. The results of the monograph may be useful for numerical integration of Ito stochastic differential equations.