39,206 research outputs found
An Integral geometry based method for fast form-factor computation
Monte Carlo techniques have been widely used in rendering algorithms for local integration. For example, to
compute the contribution of a patch to the luminance of another. In the present paper we propose an
algorithm based on Integral geometry where Monte Carlo is applied globally. We give some results of the
implementation to validate the proposition and we study the error of the technique, as well as its complexity.Postprint (published version
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
We study the approximation of by a Monte Carlo algorithm,
where is the solution of a stochastic differential equation and is a
given function. We introduce a new variance reduction method, which can be
viewed as a statistical analogue of Romberg extrapolation method. Namely, we
use two Euler schemes with steps and . This
leads to an algorithm which, for a given level of the statistical error, has a
complexity significantly lower than the complexity of the standard Monte Carlo
method. We analyze the asymptotic error of this algorithm in the context of
general (possibly degenerate) diffusions. In order to find the optimal
(which turns out to be ), we establish a central limit type theorem,
based on a result of Jacod and Protter for the asymptotic distribution of the
error in the Euler scheme. We test our method on various examples. In
particular, we adapt it to Asian options. In this setting, we have a CLT and,
as a by-product, an explicit expansion of the discretization error.Comment: Published at http://dx.doi.org/10.1214/105051605000000511 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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