勒让德多项式
数学
迭代函数
泰勒级数
傅里叶级数
随机微分方程
数学分析
傅里叶变换
系列(地层学)
维纳过程
扩展系列
应用数学
生物
古生物学
出处
期刊:Cornell University - arXiv
日期:2018-01-01
标识
DOI:10.48550/arxiv.1801.00231
摘要
The article is devoted to the practical material on expansions and mean-square approximations of specific iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 with respect to components of the multidimensional Wiener process on the base of the method of generalized multiple Fourier series. More precisely, we used the multiple Fourier--Legendre series converging in the sense of norm in the space $L_2([t, T]^k)$ $(k=1,\ldots,6)$ for approximation of iterated Ito and Stratonovich stochastic integrals. The considered iterated Ito and Stratonovich stochastic integrals are part of the stochastic Taylor expansions (Taylor-Ito and Taylor-Stratonovich expansions). Therefore, the results of the article can be useful for construction of the high-order strong numerical methods for Ito stochastic differential equations. Expansions of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 6 using Legendre polynomials are derived. The convergence with probability 1 of the mentioned method of generalized multiple Fourier series is proved for iterated Ito stochastic integrals of multiplicities $k$ $(k\in\mathbb{N})$ for the cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series.
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