相依拉丁超立方抽样的中心极限定理及其在一篮子期权定价中的应用
A central limit theorem for Latin hypercube sampling with dependence and application to exotic basket option pricing
摘要：
我们考虑估计$\mathbb{E} [f(U^1, \ldots, U^d)]$的问题，其中$(U^1, \ldots, U^d)$表示具有均匀分布边缘的随机向量。拉丁超立方抽样（LHS）是解决这类高维数值积分问题的有力工具。在随机向量$(U^1, \ldots, U^d)$的相关分量的情况下，可以通过使用具有相关性的拉丁超立方体采样（LHSD）来实现更精确的结果。本文给出了$d$维LHSD估计的中心极限定理，推广了Packham和Schmidt的一个结果.此外，我们给出了条件的功能$f$和分布的$(U^1, \ldots, U^d)$下，可以实现减少的方差。最后，我们比较了蒙特卡罗和LHSD估计的有效性在异国篮子期权定价问题的数值。
英文摘要：
We consider the problem of estimating $\mathbb{E} [f(U^1, \ldots, U^d)]$, where $(U^1, \ldots, U^d)$ denotes a random vector with uniformly distributed marginals. In general, Latin hypercube sampling (LHS) is a powerful tool for solving this kind of high-dimensional numerical integration problem. In the case of dependent components of the random vector $(U^1, \ldots, U^d)$ one can achieve more accurate results by using Latin hypercube sampling with dependence (LHSD). We state a central limit theorem for the $d$-dimensional LHSD estimator, by this means generalising a result of Packham and Schmidt. Furthermore we give conditions on the function $f$ and the distribution of $(U^1, \ldots, U^d)$ under which a reduction of variance can be achieved. Finally we compare the effectiveness of Monte Carlo and LHSD estimators numerically in exotic basket option pricing problems.
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