Multilevel Monte Carlo by using the Halton sequence

Monte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

A computational investigation of the optimal Halton sequence in QMC applications

We propose the use of randomized (scrambled) quasirandom sequences for the purpose of providing practical error estimates for quasi-Monte Carlo (QMC) applications. One popular quasirandom sequence among practitioners is the Halton sequence. However, Halton subsequences have correlation problems in their highest dimensions, and so using this sequence for high-dimensional integrals dramatically affects the accuracy of QMC. Consequently, QMC studies have previously proposed several scrambling methods; however, to varying degrees, scrambled versions of Halton sequences still suffer from the correlation problem as manifested in two-dimensional projections. This paper proposes a modified Halton sequence (MHalton), created using a linear digital scrambling method, which finds the optimal multiplier for the Halton sequence in the linear scrambling space. In order to generate better uniformity of distributed sequences, we have chosen strong MHalton multipliers up to 360 dimensions. The proposed multipliers have been tested and proved to be stronger than several sets of multipliers used in other known scrambling methods. To compare the quality of our proposed scrambled MHalton sequences with others, we have performed several extensive computational tests that use {L_{2}} -discrepancy and high-dimensional integration tests. Moreover, we have tested MHalton sequences on Mortgage-backed security (MBS), which is one of the most widely used applications in finance. We have tested our proposed MHalton sequence numerically and empirically, and they show optimal results in QMC applications. These confirm the efficiency and safety of our proposed MHalton over scrambling sequences previously used in QMC applications.

Tractability of Multivariate Integration Using Low-Discrepancy Sequences

We propose a notion of (t, e, s)-sequences in multiple bases, which unifies the Halton sequence and (t, s)-sequences under one roof, and obtain an upper bound of their discrepancy consisting only of the leading term. By using this upper bound, we improve the tractability results currently known for the Halton sequence, the Niederreiter sequence, the Sobol’ sequence, and the generalized Faure sequence, and also give tractability results for the Xing-Niederreiter sequence and the Hofer-Niederreiter sequence, for which no results have been known so far.