2013_Book_MonteCarloAndQuasi-MonteCarloM.pdf - Springer Proceedings in Mathematics Statistics Josef Dick Frances Y Kuo Gareth W Peters Ian H Sloan

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Unformatted text preview: Springer Proceedings in Mathematics & Statistics Josef Dick Frances Y. Kuo Gareth W. Peters Ian H. Sloan Editors Monte Carlo and Quasi-Monte Carlo Methods 2012 Springer Proceedings in Mathematics and Statistics Volume 65 For further volumes: Springer Proceedings in Mathematics and Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. Josef Dick • Frances Y. Kuo • Gareth W. Peters Ian H. Sloan Editors Monte Carlo and Quasi-Monte Carlo Methods 2012 123 Editors Josef Dick Frances Y. Kuo Gareth W. Peters Ian H. Sloan The University of New South Wales School of Mathematics and Statistics New South Wales Sydney, Australia ISSN 2194-1009 ISSN 2194-1017 (electronic) ISBN 978-3-642-41094-9 ISBN 978-3-642-41095-6 (eBook) DOI 10.1007/978-3-642-41095-6 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013956514 Mathematics Subject Classification (2010): Primary: 11K38, 11K45, 65-06, 65C05, 65C10, 65D30 Secondary: 11K38, 65D18, 65D32, 65R20, 91B25 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media ( ) Preface This volume represents the refereed proceedings of the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, which was held at the University of New South Wales, Sydney, Australia, from 13 to 17 February 2012. It contains a limited selection of articles based on presentations given at the conference. The conference program was arranged with the help of an international committee consisting of: • • • • • • • • • • • • • • • • • • • • • • • • • • William Chen, Macquarie University, Australia Ronald Cools, KU Leuven, Belgium Josef Dick, University of New South Wales, Australia (Conference organizer) Henri Faure, CNRS Marseille, France Alan Genz, Washington State University, USA Mike Giles, University of Oxford, UK Paul Glasserman, Columbia University, USA Michael Gnewuch, University of Kaiserslautern, Germany Stefan Heinrich, University of Kaiserslautern, Germany Fred J. Hickernell, Illinois Institute of Technology, USA Aicke Hinrichs, University of Rostock, Germany Stephen Joe, University of Waikato, New Zealand Aneta Karaivanova, Bulgarian Academy of Science, Bulgaria Alexander Keller, NVIDIA, Germany Dirk P. Kroese, University of Queensland, Australia Frances Y. Kuo, University of New South Wales, Australia (Conference organizer) Gerhard Larcher, Johannes Kepler University Linz, Austria Pierre L’Ecuyer, Université de Montréal, Canada Christiane Lemieux, University of Waterloo, Canada Peter Mathé, Weierstrass Institute Berlin, Germany Makoto Matsumoto, Hiroshima University, Japan Kerrie Mengersen, Queensland University of Technology, Australia Thomas Müller-Gronbach, University of Passau, Germany Harald Niederreiter, RICAM Linz and University of Salzburg, Austria Erich Novak, University of Jena, Germany Art B. Owen, Stanford University, USA v vi Preface • Gareth W. Peters, University of New South Wales, Australia, and University College London, UK (Conference organizer) • Friedrich Pillichshammer, Johannes Kepler University Linz, Austria • Leszek Plaskota, University of Warsaw, Poland • Eckhard Platen, University of Technology Sydney, Australia • Klaus Ritter, University of Kaiserslautern, Germany • Gareth Roberts, University of Warwick, UK • Wolfgang Ch. Schmid, University of Salzburg, Austria • Nikolai Simonov, Russian Academy of Sciences, Russia • Ian H. Sloan, University of New South Wales, Australia (Conference organizer) • Ilya M. Sobol’, Russian Academy of Sciences, Russia • Jerome Spanier, Claremont, California, USA • Shu Tezuka, Kyushu University, Japan • Xiaoqun Wang, Tsinghua University, China • Grzegorz W. Wasilkowski, University of Kentucky, USA • Henryk Wo´zniakowski, Columbia University, USA, and University of Warsaw, Poland This conference continued the tradition of biennial MCQMC conferences initiated by Harald Niederreiter, held previously at: • • • • • • • • • University of Nevada in Las Vegas, Nevada, USA, in June 1994 University of Salzburg, Austria, in July 1996 Claremont Colleges in Claremont, California, USA, in June 1998 Hong Kong Baptist University in Hong Kong, China, in November 2000 National University of Singapore, Republic of Singapore, in November 2002 Palais des Congrès in Juan-les-Pins, France, in June 2004 Ulm University, Germany, in July 2006 Université de Montréal, Canada, in July 2008 University of Warsaw, Poland, in August 2010 The next conference will be held at the KU Leuven, Belgium, in April 2014. The proceedings of these previous conferences were all published by SpringerVerlag, under the following titles: • Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P.J.-S. Shiue, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 2000 (K.-T. Fang, F.J. Hickernell, and H. Niederreiter, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 2002 (H. Niederreiter, ed.) • Monte Carlo and Quasi-Monte Carlo Methods 2004 (H. Niederreiter and D. Talay, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 2006 (A. Keller, S. Heinrich, and H. Niederreiter, eds.) Preface vii • Monte Carlo and Quasi-Monte Carlo Methods 2008 (P. L’Ecuyer and A. Owen, eds.) • Monte Carlo and Quasi-Monte Carlo Methods 2010 (L. Plaskota and H. Wo´zniakowski, eds.) The program of the conference was rich and varied with over 140 talks being presented. Highlights were the invited plenary talks given by Pierre Del Moral (INRIA and University of Bordeaux 1), Mike Giles (Oxford University), Fred J. Hickernell (Illinois Institute of Technology), Aicke Hinrichs (University of Jena), Michael Lacey (Georgia Institute of Technology), Kerrie Mengersen (Queensland University of Technology), Andreas Neuenkirch (University of Kaiserslautern), Art B. Owen (Stanford University), Leszek Plaskota (University of Warsaw), and Eckhard Platen (University of Technology Sydney), and the tutorials given by Art B. Owen (Stanford University), Pierre Del Moral (INRIA and University of Bordeaux 1), Josef Dick (University of New South Wales), and Alex Keller (NVIDIA). The papers in this volume were carefully screened and cover both the theory and the applications of Monte Carlo and quasi-Monte Carlo methods. We thank the anonymous reviewers for their reports and many others who contributed enormously to the excellent quality of the conference presentations and to the high standards for publication in these proceedings by careful review of the abstracts and manuscripts that were submitted. We gratefully acknowledge generous financial support of the conference by the School of Mathematics and Statistics of the University of New South Wales, the Australian Mathematical Society (AustMS), the Australian and New Zealand Industrial and Applied Mathematics (ANZIAM), the Australian Mathematical Sciences Institute (AMSI), the Commonwealth Scientific and Industrial Research Organisation (CSIRO), and the National Science Foundation (NSF). Finally, we want to express our gratitude to Springer-Verlag for publishing this volume. Sydney, Australia September 2013 Josef Dick Frances Y. Kuo Gareth W. Peters Ian H. Sloan Contents Part I Invited Articles Computing Functionals of Square Root and Wishart Processes Under the Benchmark Approach via Exact Simulation .. . . . . . . . . . . . . . . . . . . . Jan Baldeaux and Eckhard Platen 3 The Supremum Norm of the Discrepancy Function: Recent Results and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Dmitriy Bilyk and Michael Lacey 23 An Introduction to Stochastic Particle Integration Methods: With Applications to Risk and Insurance . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Pierre Del Moral, Gareth W. Peters, and Christelle Vergé 39 Multilevel Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Michael B. Giles 83 Guaranteed Conservative Fixed Width Confidence Intervals via Monte Carlo Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 Fred J. Hickernell, Lan Jiang, Yuewei Liu, and Art B. Owen Discrepancy, Integration and Tractability.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 Aicke Hinrichs Noisy Information: Optimality, Complexity, Tractability . . . . . . . . . . . . . . . . . . . 173 Leszek Plaskota Part II Tutorial Quasi-Monte Carlo Image Synthesis in a Nutshell . . . . . . .. . . . . . . . . . . . . . . . . . . . 213 Alexander Keller ix x Part III Contents Contributed Articles Conditional Sampling for Barrier Option Pricing Under the Heston Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253 Nico Achtsis, Ronald Cools, and Dirk Nuyens Probabilistic Star Discrepancy Bounds for Double Infinite Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 271 Christoph Aistleitner and Markus Weimar The L2 Discrepancy of Irrational Lattices . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 289 Dmitriy Bilyk Complexity of Banach Space Valued and Parametric Integration . . . . . . . . . 297 Thomas Daun and Stefan Heinrich Extended Latin Hypercube Sampling for Integration and Simulation . . . . 317 Rami El Haddad, Rana Fakhereddine, Christian Lécot, and Gopalakrishnan Venkiteswaran A Kernel-Based Collocation Method for Elliptic Partial Differential Equations With Random Coefficients . . . . . . .. . . . . . . . . . . . . . . . . . . . 331 Gregory E. Fasshauer and Qi Ye Polynomial Accelerated MCMC and Other Sampling Algorithms Inspired by Computational Optimization .. .. . . . . . . . . . . . . . . . . . . . 349 Colin Fox Antithetic Multilevel Monte Carlo Estimation for Multidimensional SDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 367 Michael B. Giles and Lukasz Szpruch On the Convergence of Quantum and Sequential Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 385 François Giraud and Pierre Del Moral Lower Error Bounds for Randomized Multilevel and Changing Dimension Algorithms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 399 Michael Gnewuch A Non-empirical Test on the Second to the Sixth Least Significant Bits of Pseudorandom Number Generators .. . . . . . . . . . . . . . . . . . . . 417 Hiroshi Haramoto, Makoto Matsumoto, Takuji Nishimura, and Yuki Otsuka A Finite-Row Scrambling of Niederreiter Sequences . . . .. . . . . . . . . . . . . . . . . . . . 427 Roswitha Hofer and Gottlieb Pirsic Contents xi Reconstructing Multivariate Trigonometric Polynomials by Sampling Along Generated Sets . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 439 Lutz Kämmerer Bayesian Approaches to the Design of Markov Chain Monte Carlo Samplers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 455 Jonathan M. Keith and Christian M. Davey Deterministic Consistent Density Estimation for Light Transport Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 467 Alexander Keller and Nikolaus Binder On Wavelet-Galerkin Methods for Semilinear Parabolic Equations with Additive Noise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 481 Mihály Kovács, Stig Larsson, and Karsten Urban Component-by-Component Construction of Hybrid Point Sets Based on Hammersley and Lattice Point Sets . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 501 Peter Kritzer, Gunther Leobacher, and Friedrich Pillichshammer A QMC-Spectral Method for Elliptic PDEs with Random Coefficients on the Unit Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 517 Quoc Thong Le Gia Sampling and Low-Rank Tensor Approximation of the Response Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 535 Alexander Litvinenko, Hermann G. Matthies, and Tarek A. El-Moselhy The Stochastic EM Algorithm for Censored Mixed Models . . . . . . . . . . . . . . . . 553 Ian C. Marschner Existence of Higher Order Convergent Quasi-Monte Carlo Rules via Walsh Figure of Merit .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 569 Makoto Matsumoto and Takehito Yoshiki ANOVA Decomposition of Convex Piecewise Linear Functions . . . . . . . . . . . . 581 Werner Römisch Hit-and-Run for Numerical Integration .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 597 Daniel Rudolf QMC Galerkin Discretization of Parametric Operator Equations . . . . . . . . . 613 Christoph Schwab On the Choice of Weights in a Function Space for Quasi-Monte Carlo Methods for a Class of Generalised Response Models in Statistics ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 631 Vasile Sinescu, Frances Y. Kuo, and Ian H. Sloan xii Contents Multi-level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems . . . . .. . . . . . . . . . . . . . . . . . . . 649 Jonas Šukys, Siddhartha Mishra, and Christoph Schwab Conference Participants .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 667 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 685 Part I Invited Articles Computing Functionals of Square Root and Wishart Processes Under the Benchmark Approach via Exact Simulation Jan Baldeaux and Eckhard Platen Abstract The aim of the paper is to show how Wishart processes can be used flexibly in financial modeling. We explain how functionals, resulting from the benchmark approach to finance, can be accurately computed via exact simulation methods. We employ Lie symmetry methods to identify explicit transition densities and explicitly computable functionals. We illustrate the proposed methods via finance problems formulated under the benchmark approach. This approach allows us to exploit conveniently the analytical tractability of the considered diffusion processes. 1 Introduction In mathematical finance, the pricing of financial derivatives can under suitable conditions be shown to amount to the computation of an expected value, see e.g. [50, 53]. We focus in this paper on the application of the benchmark approach, described e.g. in [53], where we show how Wishart processes can be flexibly used in financial modeling and derivative pricing. Depending on the financial derivative and the model under consideration, it might not be possible to compute the expected value explicitly, however, numerical methods have to be invoked. A candidate for the computation of such expectations is the Monte Carlo method, see e.g. [11, 29], J. Baldeaux () Finance Discipline Group, University of Technology, PO Box 123, Broadway, Sydney, NSW, 2007, Australia Current address: Quant Models & Development, Danske Bank, Denmark e-mail: [email protected] E. Platen Finance Discipline Group and School of Mathematical Sciences, University of Technology, PO Box 123, Broadway, Sydney, NSW, 2007, Australia e-mail: [email protected] J. Dick et al. (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Mathematics & Statistics 65, DOI 10.1007/978-3-642-41095-6__1, © Springer-Verlag Berlin Heidelberg 2013 3 4 J. Baldeaux and E. Platen and [40]. Applying the Monte Carlo method typically entails the sampling of the distribution of the relevant financial state variables, e.g. an equity index, a short rate, or a commodity price. It is then, of course, desirable to have at one’s disposal a recipe for drawing samples from the relevant distributions. In case these distributions are known, one refers to exact simulation schemes, see e.g. [52], but also [7–9], and [16], for further references on exact simulation schemes. In particular, exact simulation is relevant for long term simulation. If exact simulation schemes are not applicable, discrete time approximations, as analyzed in [40] and [52] become relevant. For modeling financial quantities of interest, it is important to know a priori if exact simulation schemes exist, so that financial derivatives can be priced accurately, even if expected values cannot be computed explicitly. In this paper, we discuss classes of square root and Wishart processes for which exact simulation is possible. For one-dimensional diffusions, Lie symmetry analysis, see [10], and [51] turns out to be a useful tool to identify tractable diffusion processes. Besides allowing one to discover transition densities, see [21], it also allows us to compute Laplace transforms of important multidimensional functionals, see e.g. [20]. In particular, squared Bessel processes fall in...
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  • Fall '11
  • Hurvich
  • Monte Carlo method, Monte Carlo methods in finance, Quasi-Monte Carlo method, Wishart Processes, Jan Baldeaux

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