Shu Tezuka

Efficient and portable combined Tausworthe random number generators

In this paper, we propose three combined Tausworthe random number generators with period length about 1018, whose k-distribution properties are good and which can be implemented in a portable way. These generators are found through an exhaustive search for the combination with the best lattice structure in GF{2, x }~, the k-dimensional vector space over the field of all Laurent series with coefficients in GF(2). We then apply a battery of statistical tests to these generators for the comprehensive investigation of their empirical statistical properties. No apparent defect was found. In the appendix, we give a sample program in C for the generators.

Polynomial arithmetic analogue of Halton sequences

introduce the radical inverse function with respect to polynomial arithmetic over finite fields,and thereby construct a polynomial arithmetic analogue of Halton sequences. Next, we general-ize the definition of Niederreiter sequences so that his key lemma [Niederreiter 1988, Lemma 4]still holds. We then prove that our analogous Halton sequences constitute a new class oflow-discrepancy sequences viewed as a special subclass of the generalized Niederreiter se-quences.Categories and Subject Descriptors: G.1.4

I -binomial scrambling of digital nets and sequences

The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owens random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions.This paper considers a reduction of randomness required for Owens random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owens key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i- binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owens scrambling also hold with the left i-binomial scrambling.

Another Random Scrambling of Digital ( t , s )-Sequences

This paper presents a new random scrambling of digital (t,s)-sequences and its application to two problems from finance, showing the usefulness of this new class of randomized low-discrepancy sequences; moreover the simplicity of the construction allows efficient implementation and should facilitate the derandomization in this particular class; also the search of the effective dimension in high dimensional applications should be improved by the use of such scramblings.

On the lattice structure of the add-with-carry and subtract-with-borrow random number generators

Marsaglia and Zaman recently proposed new classes of random number generators, called add-with-carry(AWC) and subtract-with-borrow(SWB), which are capable of quickly generating very long-period (pseudo)-random number sequences using very little memory. We show that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli. So, the AWC/SWB generators can be viewed as efficient ways of implementing such large linear congruential generators. As a consequence, the theoretical properties of such generators can be studied in the same way as for linear congruential generators, namely, via the spectral and lattice tests. We also show how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences. Our numerical examples illustrate the fact that AWC/SWB generators have extremely bad lattice structure in high dimensions.

On the distribution of

The lattice structure of conventional linear congruential random number generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F2. The state of the generator (a Laurent series) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distributed in the unit hypercube. More specifically, for any partition of the k-dimensional hypercube into 2kl identical subcubes, we can quickly compute a table giving the exact number of subcubes that contain exactly n points, for each integer n. We give numerical examples and discuss the practical implications of our results.

k

Since Black and Scholes [2] applied a stochastic process of geometric Brownian motion to modeling the evolution of stock prices, and successfully developed an elegant mathematical theory for option pricing, financial modeling based on stochastic differential equations has become the theoretical underpinning for the practice of trading financial derivatives. Why theoretical prices derived from such stochastic models are so important in practice can be explained as follows: (1) Financial institutions need to know the theoretical price when engineering “new” financial products or when mark-tomarketing some nonliquid assets, because the market price of the product has never been observed before or because the assets have not been traded lately in the actual financial market; (2) They conduct a benchmark for comparing the theoretical price with the actual price. If the difference is not negligible, it implies an arbitrage profit opportunity or mispricing.

-dimensional vectors for simple and combined Tausworthe sequences

Authors’ address: IBM Research, Tokyo Research Laboratory, 1623-14 Shimotsuruma, Yamoto, Kanagawa 242, Japan; email: {tezukaj ttoku}@trlvm.vnet.ibm.com. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and\or specific permission. @ 1994 ACM 1049-3301/94/0700-0279

Financial Applications of Monte Carlo and Quasi-Monte Carlo Methods

03.50

A note on polynomial arithmetic analogue of Halton sequences

Due to reduced profitability, increased price competition, and strengthened regulation, financial institutions in all countries are now upgrading their financial analytics based on Monte Carlo simulation. In this article, we propose three key technologies, i.e., data protection, integrity, and deadline scheduling, which are indispensable to build a secure PC-grid for financial risk management. We constructed a PC-grid by scavenging unused CPU cycles of about 50 PCs under real office environment, and obtained the 80 times speed-up, namely, for 100,000 Monte Carlo scenarios, 95 h computation on a single server is reduced to 70 min. Finally, we discuss future research directions.