Expansion of Iterated Ito Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Multiple Fourier Series Converging in the Mean

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The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple Fourier series for expansion and mean-square approximation of iterated Ito stochastic integrals of arbitrary multiplicity $k$ ($k\in\mathbb{N}$) with respect to components of the multidimensional Wiener process is proposed and developed.

• The obtained expansions contain only one operation of the limit transition in contrast to its existing analogues.
• In the article it is also obtained the generalization of the proposed method for an arbitrary complete orthonormal systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$ as well as for complete orthonormal with weight $r(t_1)\ldots r(t_k)$ systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$.
• The comparison of the considered method with the well-known expansions of iterated Ito stochastic integrals based on the Ito formula and Hermite polynomials is given.