Constructions of Low Discrepancy Sequences for Computation of Derivative Pricing

在金融工程中计算期权衍生产品的价格时，需要对其进行高维积分。对于这种具有序数伪随机数的计算问题，我们通常采用蒙特卡罗方法。(t，m，S)-网或(t，S)-序列是一类在精度方面优于随机数的平衡点集，我们研究了这种平衡点集的构造和分析。当强度t=3，4时，我们证明了一个(t，m，S)-网等价于有限射影空间中的一个构形，称为Theta-构形。证明了(3，m，S)-网等价于正交阵，最近提出了一种称为Hickrnel序列的序列。利用有限域上的多项式定义了序列，证明了它是(t，S)-序列，最均匀分布的序列是(t，S)-序列中的(0，S)-序列。我们过去是随机选择参数的。我们找到了一种选择好参数的方法。这种方法被称为“去随机化”。此外，我们还提出了一种利用i-二项式性质的方法。分析了去随机化的性质，研究并改进了一种周期很长的有序伪随机数发生器Mersenne Twister。我们还研究了与(t，m，S)-网或(t，S)-序列构造有关的许多基本组合问题

When we compute a price of option derivative in financial engineering, an integration over very high dimension is required. We usually use Monte Carlo method for such computation problems with ordinal pseudo random numbers. It is known that a kind of balanced point set is better than random numbers with respect to precisions, called (t,m,s)-nets or (t,s)-sequences.We have studied to construct and analyze such point sets. When the strength t=3, 4, we proved a (t,m,s)-net is equivalent to a configuration, called Theta-configuration, in a finite projective space. Also we proved that a (3,m,s)-net is equivalent to an orthogonal array.There are sequences called Hickrnel sequences which are proposed recently. We define the sequences using polynomials over a finite field and proved it is a (t, s)-sequence.The most uniformly distributed sequences are (0,s)-sequences in (t,s)-sequences. We used to choose parameters randomly. We found a method to choose good parameters. This method is called"De-randomization". Also we proposed an approach using i-binomial property. We analyzed properties of de-randomizations.We studied and improved an ordinal pseudo random number generator, called Mersenne twister, which have very long period. And also we studied many basic combinatorial problems which may have a relation to constructions of (t,m,s)-nets or (t,s)-sequences