空间低差异序列、球锥差异及其在金融建模中的应用
Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling
AI总结：
本文提出一种定义在R^{d+1}上的再生核希尔伯特空间，用于金融建模中的积分计算，通过扩展Stolarsky不变原理并测试其在期权定价中的有效性。
摘要：
在本文中，我们介绍了一种定义在R^{d+1}上的再生核希尔伯特空间，该空间是定义在单位球S^d上的再生核与定义在[0,∞)上的再生核的张量积。我们扩展了Stolarsky的不变原理，并在相应的再生核希尔伯特空间中证明了数值积分的上界和下界。将方向与距离从原点分离的概念也可用于构造求积方法。通过扩展保持面积的Lambert变换，利用将Sobol'点从[0,1)^d提升到球体上的方法生成球体上的点。每个Sobol'点的第d个分量经过适当变换后提供距离信息，使得生成的点集在R^d中呈正态分布。数值测试提供了在这些空间中构造类似准蒙特卡洛方法进行积分的有用性证据。我们还测试了该方法在金融应用（期权定价问题）中的效果，并将其结果与传统方法在R^d中的数值积分结果进行比较。
英文摘要：
In this paper we introduce a reproducing kernel Hilbert space defined on $\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel defined on $[0,\infty)$. We extend Stolarsky's invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space. The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on $\mathbb{S}^{d-1}$ via lifting Sobol' points in $[0,1)^{d}$ to the sphere. The $d$-th component of each Sobol' point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in $\mathbb{R}^{d}$. Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in $\mathbb{R}^{d}$.
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