五、六、七、八重迭代斯特劳尼奇随机积分的展开基于广义多重傅里叶级数
Expansion of Iterated Stratonovich Stochastic Integrals of Fifth, Sixth, Seventh and Eighth Multiplicities Based on Generalized Multiple Fourier Series
AI总结：
本文基于广义多重傅里叶级数方法，研究五至八重迭代斯特劳尼奇随机积分的展开，揭示其比伊藤积分展开更简洁，并应用于随机微分方程的数值积分。
摘要：
本文致力于基于广义多重傅里叶级数方法，构建五、六、七、八重迭代斯特劳尼奇随机积分的展开，该方法在希尔伯特空间$ L_2([t, T]^k) $中以范数收敛。具体来说，我们主要使用多重傅里叶-勒让德级数和多重三角傅里叶级数（k=1,…,8）。对于k=1,…,6，也考虑了在$ L_2([t, T]) $中任意完备正交函数系的广义多重傅里叶级数情况。最近，已经获得了k∈ℕ的迭代斯特劳尼奇随机积分的展开（定理42、44），但附加了一个额外条件。所考虑的展开在均方意义上收敛，并且仅包含一次极限过渡操作，不同于现有类似展开。迭代斯特劳尼奇随机积分的展开比相应的迭代伊藤随机积分展开要简单得多。我们使用后者展开作为证明前者的工具。迭代斯特劳尼奇随机积分是伊藤随机微分方程解的泰勒-斯特劳尼奇展开的一部分。因此，本文的结果可以应用于伊藤随机微分方程的数值积分。
英文摘要：
The article is devoted to the construction of expansions of iterated Stratonovich stochastic integrals of fifth, sixth, seventh and eighth 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 $(k=1,\ldots,8)$. The case of generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in $L_2([t, T])$ is also considered for $k=1,\ldots,6$. 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.
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