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Infinite-Dimensional Quadrature and Approximation of Distributions

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Abstract

We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

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References

  1. A. Ayache, M.S. Taqqu, Rate optimality of wavelet series approximations of fractional Brownian motion, J. Fourier Anal. Appl. 9, 451–471 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  2. N.S. Bakhvalov, On approximate computation of integrals, Vestnik MGV, Ser. Math. Mech. Astron. Phys. Chem. 4, 3–18 (1959) (in Russian).

    Google Scholar 

  3. N.S. Bakhvalov, On the optimality of linear methods for operator approximation in convex classes of functions, USSR Comput. Math. Math. Phys. 11, 244–249 (1971).

    Article  Google Scholar 

  4. V. Bally, D. Talay, The law of the Euler scheme for stochastic differential equations, I: Convergence rate of the distribution function, Probab. Theory Relat. Fields 104, 43–60 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Belinsky, W. Linde, Small ball probabilities of fractional Brownian sheets via fractional integration operators, J. Theor. Probab. 15, 589–612 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  6. N. Bouleau, D. Lépingle, Numerical Methods for Stochastic Processes (Wiley, New York, 1994).

    MATH  Google Scholar 

  7. G. Chen, G. Fang, Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure, J. Complex. 20, 858–875 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  8. E. Csáki, On small values of the square integral of a multiparameter Wiener process, in Statistics and Probability, ed. by J. Mogyorodi, I. Vincze, W. Wertz (Reidel, Dordrecht, 1984), pp. 19–26.

    Google Scholar 

  9. J. Creutzig, Approximation of Gaussian random vectors in Banach spaces, Ph.D. Dissertation, Universität Jena, 2002.

  10. S. Dereich, High resolution coding of stochastic processes and small ball probabilities. Ph.D. Dissertation, TU Berlin, 2003.

  11. S. Dereich, The coding complexity of diffusion processes under supremum norm distortion, Stoch. Process. Appl. 118(6), 917–937 (2007). doi:10.1016/j.spa.2007.07.003.

    Article  MathSciNet  Google Scholar 

  12. S. Dereich, The coding complexity of diffusion processes under L p[0,1]-norm distortion, Stoch. Process. Appl. 118(6), 938–951 (2007). doi:10.1016/j.spa.2007.07.002.

    Article  MathSciNet  Google Scholar 

  13. S. Dereich, M. Scheutzow, High-resolution quantization and entropy coding for fractional Brownian motion, Electron. J. Probab. 11, 700–722 (2005).

    MathSciNet  Google Scholar 

  14. S. Dereich, F. Fehringer, A. Matoussi, M. Scheutzow, On the link between small ball probabilities and the quantization problem, J. Theor. Probab. 16, 249–265 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Dzhaparidze, H. van Zanten, Optimality of an explicit series expansion of the fractional Brownian sheet, Stat. Probab. Lett. 71, 295–301 (2005).

    Article  MATH  Google Scholar 

  16. J.A. Fill, F. Torcaso, Asymptotic analysis via Mellin transforms for small deviations in L 2-norm of integrated Brownian sheets, Probab. Theory Relat. Fields 130, 259–288 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  17. M.B. Giles, Multi-level Monte Carlo path simulation, Report NA-06/03, Oxford Univ. Computing Lab. Oper. Res. (2006, to appear).

  18. S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math., vol. 1730 (Springer, Berlin, 2000).

    MATH  Google Scholar 

  19. R.M. Gray, D.L. Neuhoff, P.C. Shields, A generalization of Ornstein’s \(\overline{d}\) distance with applications to information theory, Ann. Appl. Probab. 3, 315–328 (1975).

    MATH  MathSciNet  Google Scholar 

  20. S. Heinrich, Monte Carlo complexity of global solution of integral equations, J. Complex. 14, 151–175 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Heinrich, Multilevel Monte Carlo methods, in Large Scale Scientific Computing, ed. by S. Margenov, J. Wasniewski, P. Yalamov, Lecture Notes in Comp. Sci., vol. 2179 (Springer, Berlin, 2001), pp. 58–67.

    Chapter  Google Scholar 

  22. S. Heinrich, E. Sindambiwe, Monte Carlo complexity of parametric integration, J. Complex. 15, 317–341 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  23. J.-P. Kahane, Some Random Series of Functions (Cambridge Univ. Press, Cambridge, 1993).

    MATH  Google Scholar 

  24. L.V. Kantorovich, G.S. Rubinstein, On a space of completely additive functions, Vestnik Leningrad Univ. Ser. Mat. Astron. Phys. 2 13(7), 52–59 (1958) (in Russian).

    Google Scholar 

  25. A. Kebaier, Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing, Ann. Appl. Probab. 15, 2681–2705 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Kühn, W. Linde, Optimal series representation of fractional Brownian sheets, Bernoulli 8, 669–696 (2002).

    MATH  MathSciNet  Google Scholar 

  27. J. Kuelbs, W.V. Li, Shao, Q.M., Small ball estimates for fractional Brownian motion under Hölder norm and Chung’s functional LIL, J. Theor. Probab. 8, 361–386 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  28. W.V. Li, Q.-M. Shao, Small ball estimates for Gaussian processes under the Sobolev norm, J. Theor. Probab. 12, 699–720 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  29. W.V. Li, Q.-M. Shao, Gaussian processes: Inequalities, small ball probabilities and applications, in Stochastic Processes: Theory and Methods, ed. by D.N. Shanbhag, C.R. Rao, Handbook of Statist., vol. 19 (North-Holland, Amsterdam, 2001), pp. 533–597.

    Google Scholar 

  30. H. Luschgy, G. Pagès, Sharp asymptotics of the functional quantization problem for Gaussian processes, Ann. Appl. Probab. 32, 1574–1599 (2004).

    MATH  Google Scholar 

  31. H. Luschgy, G. Pagès, Functional quantization of a class of Brownian diffusion: A constructive approach, Stoch. Process. Appl. 116, 310–336 (2006).

    Article  MATH  Google Scholar 

  32. V. Maiorov, Widths of spaces endowed with a Gaussian measure, Russ. Acad. Sci. Dokl. Math. 45, 305–309 (1992).

    MathSciNet  Google Scholar 

  33. V. Maiorov, Average n-widths of the Wiener space in the L ∞-norm, J. Complex. 9, 222–230 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  34. P. Mathé, s-numbers in information-based complexity, J. Complex. 6, 41–66 (1990).

    Article  MATH  Google Scholar 

  35. A.S. Nemirovsky, D.B. Yudin, Problem Complexity and Method Efficiency in Optimization (Wiley, New York, 1983).

    MATH  Google Scholar 

  36. E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis, Lecture Notes in Math., vol. 1349 (Springer, Berlin, 1988).

    MATH  Google Scholar 

  37. G. Pagès, J. Printems, Functional quantization for numerics with an application to option pricing, Monte Carlo Methods Appl. 11, 407–446 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  38. S.T. Rachev, Probability Metrics and the Stability of Stochastic Models (Wiley, Chichester, 1991).

    MATH  Google Scholar 

  39. K. Ritter, Average-Case Analysis of Numerical Problems, Lecture Notes in Math., vol. 1733 (Springer, Berlin, 2000).

    MATH  Google Scholar 

  40. T. Sakai, Riemannian Geometry, Trans. Math. Monogr., vol. 149 (AMS, Providence, 1996).

    MATH  Google Scholar 

  41. S.A. Smolyak, On optimal restoration of functions and functionals of them, Candidate Dissertation, Moscow State University, 1965 (in Russian).

  42. D. Talay, L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stoch. Anal. Appl. 8, 483–509 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  43. J.F. Traub, G.W. Wasilkowski, H. Woźniakowski, Information-Based Complexity (Academic, New York, 1988).

    MATH  Google Scholar 

  44. G.W. Wasilkowski, Randomization for continuous problems, J. Complex. 5, 195–218 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  45. G.W. Wasilkowski, H. Woźniakowski, On tractability of path integration, J. Math. Phys. 37, 2071–2088 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  46. G.W. Wasilkowski, H. Woźniakowski, Complexity of weighted approximation over ℝd, J. Complex. 17, 722–740 (2001).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Jakob Creutzig & Klaus Ritter

  2. Institut für Mathematik, MA 7-5, Fakultät II, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Steffen Dereich

  3. Institut für Mathematische Stochastik, Fakultät für Mathematik, Universität Magdeburg, Postfach 4120, 39016, Magdeburg, Germany

    Thomas Müller-Gronbach

Authors
  1. Jakob Creutzig
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  2. Steffen Dereich
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  3. Thomas Müller-Gronbach
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  4. Klaus Ritter
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Correspondence to Steffen Dereich.

Additional information

Communicated by Arieh Iserles.

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Creutzig, J., Dereich, S., Müller-Gronbach, T. et al. Infinite-Dimensional Quadrature and Approximation of Distributions. Found Comput Math 9, 391–429 (2009). https://doi.org/10.1007/s10208-008-9029-x

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