831 research outputs found
Piecewise Constant Martingales and Lazy Clocks
This paper discusses the possibility to find and construct \textit{piecewise
constant martingales}, that is, martingales with piecewise constant sample
paths evolving in a connected subset of . After a brief review of
standard possible techniques, we propose a construction based on the sampling
of latent martingales with \textit{lazy clocks} . These
are time-change processes staying in arrears of the true time but that
can synchronize at random times to the real clock. This specific choice makes
the resulting time-changed process a martingale
(called a \textit{lazy martingale}) without any assumptions on , and
in most cases, the lazy clock is adapted to the filtration of the lazy
martingale . This would not be the case if the stochastic clock
could be ahead of the real clock, as typically the case using standard
time-change processes. The proposed approach yields an easy way to construct
analytically tractable lazy martingales evolving on (intervals of)
.Comment: 17 pages, 8 figure
Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea
Prices of path dependent options may be modeled as expectations of functions of an infinite sequence of real variables. This talk presents recent work on bounding the error of such expectations using quasi-Monte Carlo algorithms. The expectation is approximated by an average of samples, and the functional of an infinite number of variables is approximated by a function of only variables. A multilevel algorithm employing a sum of sample averages, each with different truncated dimensions, , and different sample sizes, , yields faster convergence than a single level algorithm. This talk presents results in the worst-case error setting
06391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems
From 24.09.06 to 29.09.06, the Dagstuhl Seminar 06391 ``Algorithms and Complexity for Continuous Problems\u27\u27 was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar
are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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