Quasi-Monte Carlo Methods in Cash Flow Testing Simulations

What actuaries call cash flow testing is a large-scale simulation pitting a company\u27\u27s current policy obligation against future earnings based on interest rates. While life contingency issues associated with contract payoff are a mainstay of the actuarial sciences, modeling the random fluctuations of US Treasury rates is less studied. Furthermore, applying standard simulation techniques, such as the Monte Carlo method, to actual multi-billion dollar companies produce a simulation that can be computationally prohibitive. In practice, only hundreds of sample paths can be considered, not the usual hundreds of thousands one might expect for a simulation of this complexity. Hence, insurance companies have a desire to accelerate the convergence of the estimation procedure. The paper reports the results of cash flow testing simulations performed for Conseco L.L.C. using so-called quasi-Monte Carlo techniques. In these, pseudo-random number generation is replaced with deterministic low discrepancy sequences. It was found that by judicious choice of subsequences, that the quasi-Monte Carlo method provided a consistently tighter estimate than the traditional methods for a fixed, small number of sample paths. The techniques used to select these subsequences are discussed

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results. On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's $(t,s)$-sequences

Explicit constructions of point sets and sequences with low discrepancy

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results

Discrepancy bounds for low-dimensional point sets

The class of $(t,m,s)$-nets and $(t,s)$-sequences, introduced in their most general form by Niederreiter, are important examples of point sets and sequences that are commonly used in quasi-Monte Carlo algorithms for integration and approximation. Low-dimensional versions of $(t,m,s)$-nets and $(t,s)$-sequences, such as Hammersley point sets and van der Corput sequences, form important sub-classes, as they are interesting mathematical objects from a theoretical point of view, and simultaneously serve as examples that make it easier to understand the structural properties of $(t,m,s)$-nets and $(t,s)$-sequences in arbitrary dimension. For these reasons, a considerable number of papers have been written on the properties of low-dimensional nets and sequences