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The ANOVA decomposition of a non-smooth function of infinitely many variables can have every term smooth


Authors: Michael Griebel, Frances Y. Kuo and Ian H. Sloan
Journal: Math. Comp. 86 (2017), 1855-1876
MSC (2010): Primary 41A63, 41A99; Secondary 65D30
DOI: https://doi.org/10.1090/mcom/3171
Published electronically: October 7, 2016
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Abstract: The pricing problem for a continuous path-dependent option results in a path integral which can be recast into an infinite-dimensional integration problem. We study ANOVA decomposition of a function of infinitely many variables arising from the Brownian bridge formulation of the continuous option pricing problem. We show that all resulting ANOVA terms can be smooth in this infinite-dimensional case, despite the non-smoothness of the underlying payoff function. This result may explain why quasi-Monte Carlo methods or sparse grid quadrature techniques work for such option pricing problems.


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  • [1] Peter Acworth, Mark Broadie, and Paul Glasserman, A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing, Monte Carlo and quasi-Monte Carlo methods 1996 (Salzburg), Lecture Notes in Statist., vol. 127, Springer, New York, 1998, pp. 1-18. MR 1644509, https://doi.org/10.1007/978-1-4612-1690-2_1
  • [2] Jan Baldeaux and Michael Gnewuch, Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition, SIAM J. Numer. Anal. 52 (2014), no. 3, 1128-1155. MR 3200424, https://doi.org/10.1137/120896001
  • [3] Patrick Billingsley, Probability and Measure, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Wiley Series in Probability and Mathematical Statistics. MR 534323
  • [4] Vladimir I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. MR 1642391
  • [5] Hans-Joachim Bungartz and Michael Griebel, Sparse grids, Acta Numer. 13 (2004), 147-269. MR 2249147, https://doi.org/10.1017/S0962492904000182
  • [6] R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1, 27-46 (1997).
  • [7] Z. Ciesielski, Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403-413. MR 0132591
  • [8] M. Davis, Construction of Brownian motion, Stochastic Processes I, 2004-2005. http://www2.imperial.ac.uk/~mdavis/msc_home_page.htm
  • [9] Josef Dick and Michael Gnewuch, Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition, J. Approx. Theory 184 (2014), 111-145. MR 3218795, https://doi.org/10.1016/j.jat.2014.04.014
  • [10] Josef Dick, Frances Y. Kuo, and Ian H. Sloan, High-dimensional integration: the quasi-Monte Carlo way, Acta Numer. 22 (2013), 133-288. MR 3038697
  • [11] Paul Glasserman, Monte Carlo Methods in Financial Engineering: Stochastic Modelling and Applied Probability, Applications of Mathematics (New York), vol. 53, Springer-Verlag, New York, 2004. MR 1999614
  • [12] Michael Griebel, Frances Y. Kuo, and Ian H. Sloan, The smoothing effect of the ANOVA decomposition, J. Complexity 26 (2010), no. 5, 523-551. MR 2719646, https://doi.org/10.1016/j.jco.2010.04.003
  • [13] Michael Griebel, Frances Y. Kuo, and Ian H. Sloan, The smoothing effect of integration in $ \mathbb{R}^d$ and the ANOVA decomposition, Math. Comp. 82 (2013), no. 281, 383-400. MR 2983028, https://doi.org/10.1090/S0025-5718-2012-02578-6
  • [14] M. Griebel, F. Y. Kuo, and I. H. Sloan, Note on ``the smoothing effect of integration in $ \mathbb{R}^d$ and the ANOVA decomposition'', Math. Comp., this issue.
  • [15] Fred J. Hickernell, Thomas Müller-Gronbach, Ben Niu, and Klaus Ritter, Multi-level Monte Carlo algorithms for infinite-dimensional integration on $ \mathbb{R}^{\mathbb{N}}$, J. Complexity 26 (2010), no. 3, 229-254. MR 2657363, https://doi.org/10.1016/j.jco.2010.02.002
  • [16] Junichi Imai and Ken Seng Tan, Minimizing effective dimension using linear transformation, Monte Carlo and quasi-Monte Carlo methods 2002, Springer, Berlin, 2004, pp. 275-292. MR 2076939
  • [17] Steven G. Krantz and Harold R. Parks, The Implicit Function Theorem: History, Theory, and Applications, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1894435
  • [18] F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and H. Woźniakowski, On decompositions of multivariate functions, Math. Comp. 79 (2010), no. 270, 953-966. MR 2600550, https://doi.org/10.1090/S0025-5718-09-02319-9
  • [19] Paul Lévy, Processus Stochastiques et Mouvement Brownien. Suivi d'une note de M. Loève, Gauthier-Villars, Paris, 1948 (French). MR 0029120
  • [20] Ruixue Liu and Art B. Owen, Estimating mean dimensionality of analysis of variance decompositions, J. Amer. Statist. Assoc. 101 (2006), no. 474, 712-721. MR 2281247, https://doi.org/10.1198/016214505000001410
  • [21] I. M. Sobol, Sensitivity estimates for nonlinear mathematical models, Math. Modeling Comput. Experiment 1 (1993), no. 4, 407-414 (1995). MR 1335161
  • [22] J. Michael Steele, Stochastic Calculus and Financial Applications, Applications of Mathematics (New York), vol. 45, Springer-Verlag, New York, 2001. MR 1783083
  • [23] Hans Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, EMS Tracts in Mathematics, vol. 11, European Mathematical Society (EMS), Zürich, 2010. MR 2667814
  • [24] Hans Triebel, Numerical integration and discrepancy, a new approach, Math. Nachr. 283 (2010), no. 1, 139-159. MR 2598598, https://doi.org/10.1002/mana.200910842
  • [25] Grzegorz W. Wasilkowski and Henryk Woźniakowski, On tractability of path integration, J. Math. Phys. 37 (1996), no. 4, 2071-2086. MR 1380891, https://doi.org/10.1063/1.531493

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Additional Information

Michael Griebel
Affiliation: Institut für Numerische Simulation, Universität Bonn, Wegelerstrasse 6, 53115, Bonn, Germany – and – Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven, 53754 Sankt Augustin, Germany
Email: griebel@ins.uni-bonn.de

Frances Y. Kuo
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: f.kuo@unsw.edu.au

Ian H. Sloan
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: i.sloan@unsw.edu.au

DOI: https://doi.org/10.1090/mcom/3171
Received by editor(s): May 14, 2014
Received by editor(s) in revised form: June 23, 2015, and November 29, 2015
Published electronically: October 7, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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