论文标题
基于广义的多个傅立叶系列的迭代ITO和Stratonovich随机积分的强近似。应用于ITO SDE和半线性SPDE的数值解决方案
Strong Approximation of Iterated Ito and Stratonovich Stochastic Integrals Based on Generalized Multiple Fourier Series. Application to Numerical Solution of Ito SDEs and Semilinear SPDEs
论文作者
论文摘要
本书专门用于在ITO SDE的数值集成和具有非线性乘法性痕量类别噪声的非交通性半线性SPDE的情况下的迭代随机积分（ISI）的强近似。该专着为ISIS研究开辟了一个新的方向。我们首次成功使用$ l_2（[t，t]^k）$的常规多重傅立叶系列，用于扩展和强近似Mulusterity $ k，$ $ k \ in {\ bf n} $（第1章）的扩展和强近似。该结果已适用于1至8的Stratonovich ISI（$ l_2（[t，t]）$中的Legendre多项式或三角函数的持续微分重量函数的情况以及tratonovich ISIS的多项式或三角函数的含量为1至6（持续的次数均为$ l_ as usiviens us untough and untound and ustim untought and t $ L_对于其他一些类型的迭代随机积分（第1章）。最近（在2024年），对于$ k $ $ $ $ $ $ $ $（k \ in {\ bf n}）$的Stratonovich ISIS也进行了提及的改编，用于$ l_2（[t，t]）$（theorems 2.59，2.61）的任意缺点的情况，但在一个其他条件下。我们得出了均值$ k $，$ k \ in {\ bf n} $（第1章）的ITO ISIS近似近似的均方误差的确切和近似表达式。我们提供了一种重要的实用材料（第5章），该材料使用Legendre polynomials的cons of Thelor-Ito和Taylor-stratonovich扩展（第4章），利用Legendre polynomials和Trigonometric函数的CONS的泰勒 - ito和Taylor-Statonovich扩展（第4章）。本书中提出的方法已与一些现有方法进行了比较（第6章）。第1章的结果（第7章）用于相对于$ q $ - weener进程的{\ bf n} $的多重性$ k，$ k \ in {\ bf n} $的近似。
The book is devoted to the strong approximation of iterated stochastic integrals (ISIs) in the context of numerical integration of Ito SDEs and non-commutative semilinear SPDEs with nonlinear multiplicative trace class noise. The monograph opens up a new direction in researching of ISIs. For the first time we successfully use the generalized multiple Fourier series converging in the sense of norm in $L_2([t, T]^k)$ for the expansion and strong approximation of Ito ISIs of multiplicity $k,$ $k\in{\bf N}$ (Chapter 1). This result has been adapted for Stratonovich ISIs of multiplicities 1 to 8 (the case of continuously differentiable weight functions and a CONS of Legendre polynomials or trigonometric functions in $L_2([t, T])$) and for Stratonovich ISIs of multiplicities 1 to 6 (the case of continuous weight functions and an arbitrary CONS in $L_2([t, T])$) (Chapter 2), as well as for some other types of iterated stochastic integrals (Chapter 1). Recently (in 2024), the mentioned adaptation has also been carried out for Stratonovich ISIs of multiplicity $k$ $(k\in{\bf N})$ for the case of an arbitrary CONS in $L_2([t, T])$ (Theorems 2.59, 2.61) but under one additional condition. We derived the exact and approximate expressions for the mean-square error of approximation of Ito ISIs of multiplicity $k$, $k\in{\bf N}$ (Chapter 1). We provided a significant practical material (Chapter 5) devoted to the expansions of specific Ito and Stratonovich ISIs of multiplicities 1 to 6 from the Taylor-Ito and Taylor-Stratonovich expansions (Chapter 4) using the CONS of Legendre polynomials and the CONS of trigonometric functions. The methods formulated in this book have been compared with some existing methods (Chapter 6). The results of Chapter 1 were applied (Chapter 7) to the approximation of ISIs of multiplicity $k,$ $k\in{\bf N}$ with respect to the $Q$-Wiener process.
