22 research outputs found
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
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
discrepancy. In 1954 Roth proved a lower bound for the
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
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
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
discrepancy of finite point sets for . The
main aim of this article is to give an overview of these constructions and
related results
Gaussian limits for discrepancies. I: Asymptotic results
We consider the problem of finding, for a given quadratic measure of
non-uniformity of a set of points (such as star-discrepancy or
diaphony), the asymptotic distribution of this discrepancy for truly random
points in the limit . We then examine the circumstances under which
this distribution approaches a normal distribution. For large classes of
non-uniformity measures, a Law of Many Modes in the spirit of the Central Limit
Theorem can be derived.Comment: 25 pages, Latex, uses fleqn.sty, a4wide.sty, amsmath.st
Discrepancy bounds for low-dimensional point sets
The class of -nets and -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 -nets and
-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 -nets and
-sequences in arbitrary dimension. For these reasons, a considerable
number of papers have been written on the properties of low-dimensional nets
and sequences