Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth and Seventh Multiplicity Based on Generalized Multiple Fourier Series

The article is devoted to the construction of expansions of iterated Stratonovich stochastic integrals of fifth, sixth and seventh multiplicities based on the method of generalized multiple Fourier series converging in the sense of norm in the Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ Specifically, we mainly use multiple Fourier-Legendre series and multiple trigonometric Fourier series $(k=1,\ldots,7)$. Recently, expansions of iterated Stratonovich stochastic integrals of multiplicity $k,$ $k\in\mathbb{N}$ (the case of continuous weight functions and an arbitrary complete orthonormal system of functions in $L_2([t, T])$) have been obtained (Theorems 42, 44) but under one additional condition. The considered expansions converge in the mean-square sense and contain only one operation of the limit transition in contrast to its existing analogues. Expansions of iterated Stratonovich stochastic integrals turned out much simpler than appropriate expansions of iterated Ito stochastic integrals. We use expansions of the latter as a tool of the proof of expansions for iterated Stratonovich stochastic integrals. Iterated Stratonovich stochastic integrals are part of the Taylor-Stratonovich expansion for solutions of Ito stochastic differential equations. That is why the results of the article can be applied to the numerical integrations of Ito stochastic differential equations.