To any positive number $\varepsilon$ and any nonnegative even Schwartz function $w:\mathbb{R}\to\mathbb{R}$ we associate the random function $u^\varepsilon$ on the $m$-torus $T^m_\varepsilon:=\mathbb{R}^m/(\varepsilon^{-1}\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \sum_{ν\in\mathbb{Z}^m} X_{ν,\varepsilon} \exp\bigl(\; 2π\varepsilon \sqrt{-1} \;(ν\cdot θ)\;\bigr),$$ where $X_{ν,\varepsilon}$ are complex independent Gaussian random variables with variance $w(\varepsilon|ν|)$. Let $N^\varepsilon$ denote the number of critical points of $u^\varepsilon$. We describe explicitly two constants $C, C'$ such that as $\varepsilon$ goes to the zero, the expectation of the random variable $\frac{1}{{\rm vol}\,(T^m_\varepsilon)}N^\varepsilon$ converges to $C$, while its variance is extremely small and behaves like $C'\varepsilon^{m}$.
A new proof of the expansion of iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process based on generalized multiple Fourier series and Hermite polynomials
Multidimensional Quasi-Monte Carlo Malliavin Greeks
我们调查使用Malliavin演算，以计算多维复杂的路径依赖选项的希腊人通过模拟。为了这个目的，我们扩展的公式所采用的Kantao和Kohatsu-Higa多维的情况下。多维设置显示了Malliavin微积分方法在以前提出的不同技术上的便利性。实际上，这些技术可能在计算上是昂贵的，并且不提供用于方差缩减的灵活性。相比之下，Malliavin方法通过提供一类以不同精度返回相同期望值（希腊语）的函数来表现出更高的灵活性。如果不使用Malliavin微积分的分部积分公式，这种方差缩减的通用性是不可能的。在多维的背景下，我们找到方便的公式，允许改进的本地化技术，引入Fournié等人，并减少计算成本和方差。此外，我们表明，用于方差缩减的参数可以得到\textit{on the flight}在模拟中。我们说明了所提出的程序的效率，再加上增强版的拟蒙特卡罗模拟讨论Sabino，在一个多维的Black-Scholes市场的看涨，数字亚洲风格和异国情调的篮子期权与固定和浮动的执行价格的三角洲的数值估计。
New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series
Optimization of the Mean-Square Approximation Procedures for Iterated Ito Stochastic Integrals of Multiplicities 1 to 5 from the Unified Taylor-Ito Expansion Based on Multiple Fourier-Legendre Series
