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Pierre Etore

Bio: Pierre Etore is an academic researcher from University of Grenoble. The author has contributed to research in topics: Stochastic differential equation & Brownian motion. The author has an hindex of 7, co-authored 15 publications receiving 214 citations. Previous affiliations of Pierre Etore include École Polytechnique & Chicago Metropolitan Agency for Planning.

Papers
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TL;DR: In this article, a stratified sampling algorithm is proposed in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum.
Abstract: In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm.

52 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extended the exact simulation methods of Beskos et al. to the solutions of one-dimensional stochastic differential equations involving the local time of the unknown process at point zero.
Abstract: In this article we extend the exact simulation methods of Beskos et al to the solutions of one-dimensional stochastic differential equations involving the local time of the unknown process at point zero In order to perform the method we compute the law of the skew Brownian motion with drift The method presented in this article covers the case where the solution of the SDE with local time corresponds to a divergence form operator with a discontinuous coefficient at zero Numerical examples are shown to illustrate the method and the performances are compared with more traditional discretization schemes

51 citations

Journal ArticleDOI
TL;DR: In this paper, a stratified sampling algorithm is proposed to adaptively modify the proportion of further drawings in each stratum to converge to the optimal allocation in terms of variance reduction and the stratified estimator is asymptotically normal with a variance equal to the minimal one.
Abstract: In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction and our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm. For the pricing of arithmetic average Asian options in the Black and Scholes model, the variance is divided by a factor going from 1.1 to 50.4 (depending on the option type and the moneyness) in comparison with the standard allocation procedure, while the increase in computation time does not overcome 1%.

46 citations

Posted Content
TL;DR: In this paper, the authors investigated the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces and investigated a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within each strata.
Abstract: This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.

23 citations

Journal ArticleDOI
TL;DR: It turns out that the limiting variance depends on the directions defining the hyperrectangles but not on the precise abscissa of their boundaries along these directions, which gives a mathematical justification to the common choice of equiprobable strata.
Abstract: This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to the subsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator “on the fly”, by jointly sampling and adapting the strata which are hyperrectangles and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). It turns out that the limiting variance depends on the directions defining the hyperrectangles but not on the precise abscissa of their boundaries along these directions, which gives a mathematical justification to the common choice of equiprobable strata. So, only the directions are adaptively modified by our algorithm. We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.

23 citations


Cited by
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Book
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

1,957 citations

Book ChapterDOI
15 Apr 2013

357 citations

Book ChapterDOI
01 Jan 2013
TL;DR: This paper studies the practical issues involved in developing a real-world UIM-based performance metric and develops a prototype implementation which is used to evaluate a number of different artificial agents.
Abstract: The Universal Intelligence Measure is a recently proposed formal definition of intelligence. It is mathematically specified, extremely general, and captures the essence of many informal definitions of intelligence. It is based on Hutter’s Universal Artificial Intelligence theory, an extension of Ray Solomonoff’s pioneering work on universal induction. Since the Universal Intelligence Measure is only asymptotically computable, building a practical intelligence test from it is not straightforward. This paper studies the practical issues involved in developing a real-world UIM-based performance metric. Based on our investigation, we develop a prototype implementation which we use to evaluate a number of different artificial agents.

65 citations

Book ChapterDOI
05 Oct 2011
TL;DR: This paper describes two strategies based on pulling the arms proportionally to an upper-bound on their variance and derive regret bounds for these strategies and shows that the performance of these allocation strategies depends not only on the variances of the arms but also on the full shape of their distribution.
Abstract: In this paper, we study the problem of estimating the mean values of all the arms uniformly well in the multi-armed bandit setting. If the variances of the arms were known, one could design an optimal sampling strategy by pulling the arms proportionally to their variances. However, since the distributions are not known in advance, we need to design adaptive sampling strategies to select an arm at each round based on the previous observed samples. We describe two strategies based on pulling the arms proportionally to an upper-bound on their variance and derive regret bounds for these strategies. We show that the performance of these allocation strategies depends not only on the variances of the arms but also on the full shape of their distribution.

61 citations

Journal ArticleDOI
TL;DR: This paper proposes estimating the Shapley value using a reinforcement learning algorithm that approximates optimal stratified sampling and applies this algorithm to a DR program that utilizes the SV for payments and quantifies the accuracy of the resulting estimates.
Abstract: Designing fair compensation mechanisms for demand response (DR) is challenging. This paper models the problem in a game theoretic setting and designs a payment distribution mechanism based on the Shapley value (SV). As exact computation of the SV is in general intractable, we propose estimating it using a reinforcement learning algorithm that approximates optimal stratified sampling. We apply this algorithm to a DR program that utilizes the SV for payments and quantify the accuracy of the resulting estimates.

60 citations