Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2010, Article ID 378519, 32 pages
http://dx.doi.org/10.1155/2010/378519
Research Article

A Constructive Sharp Approach to Functional Quantization of Stochastic Processes

FB4-Department of Mathematics, University of Trier, 54286 Trier, Germany

Received 1 June 2010; Accepted 21 September 2010

Academic Editor: Peter Spreij

Copyright © 2010 Stefan Junglen and Harald Luschgy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
  2. R. M. Gray and D. L. Neuhoff, “Quantization,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2325–2383, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. V. Bally, G. Pagès, and J. Printems, “A quantization tree method for pricing and hedging multidimensional American options,” Mathematical Finance, vol. 15, no. 1, pp. 119–168, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  4. G. Pagès and J. Printems, “Optimal quadratic quantization for numerics: the Gaussian case,” Monte Carlo Methods and Applications, vol. 9, no. 2, pp. 135–165, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. Pagès and J. Printems, “Functional quantization for numerics with an application to option pricing,” Monte Carlo Methods and Applications, vol. 11, no. 4, pp. 407–446, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, vol. 1730 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.
  7. H. Luschgy and G. Pagès, “Sharp asymptotics of the functional quantization problem for Gaussian processes,” The Annals of Probability, vol. 32, no. 2, pp. 1574–1599, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Dereich, High resolution coding of stochastic processes and small ball probabilities, Ph.D. thesis, Technische Universität Berlin, 2003.
  9. S. Dereich, “The coding complexity of diffusion processes under supremum norm distortion,” Stochastic Processes and Their Applications, vol. 118, no. 6, pp. 917–937, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S. Dereich, “The coding complexity of diffusion processes under Lp[0,1]-norm distortion,” Stochastic Processes and Their Applications, vol. 118, no. 6, pp. 938–951, 2008. View at Publisher · View at Google Scholar
  11. S. Dereich and M. Scheutzow, “High-resolution quantization and entropy coding for fractional Brownian motion,” Electronic Journal of Probability, vol. 11, no. 28, pp. 700–722, 2006. View at Google Scholar · View at Zentralblatt MATH
  12. J. Creutzig, Approximation of Gaussian random vectors in Banach spaces, Ph.D. thesis, University of Jena, 2002.
  13. S. Graf, H. Luschgy, and G. Pagès, “Functional quantization and small ball probabilities for Gaussian processes,” Journal of Theoretical Probability, vol. 16, no. 4, pp. 1047–1062, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. F. Aurzada and S. Dereich, “The coding complexity of Lévy processes,” Foundations of Computational Mathematics, vol. 9, no. 3, pp. 359–390, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. F. Aurzada and S. Dereich, “Small deviations of general Lévy processes,” The Annals of Probability, vol. 37, no. 5, pp. 2066–2092, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. F. Aurzada, S. Dereich, M. Scheutzow, and C. Vormoor, “High resolution quantization and entropy coding of jump processes,” Journal of Complexity, vol. 25, no. 2, pp. 163–187, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. H. Luschgy and G. Pagès, “Functional quantization rate and mean regularity of processes with an application to Lévy processes,” The Annals of Applied Probability, vol. 18, no. 2, pp. 427–469, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. H. Luschgy, G. Pagès, and B. Wilbertz, “Asymptotically optimal quantization schemes for Gaussian processes,” http://arxiv.org/abs/0802.3761.
  19. G. Pagès, “A space quantization method for numerical integration,” Journal of Computational and Applied Mathematics, vol. 89, no. 1, pp. 1–38, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Wilbertz, Construction of optimal quantizers for Gaussian measures on Banach spaces, Ph.D. thesis, University of Trier, 2008.
  21. H. Luschgy and G. Pagès, “Expansions for Gaussian processes and Parseval frames,” Electronic Journal of Probability, vol. 14, no. 42, pp. 1198–1221, 2009. View at Google Scholar
  22. H. Luschgy and G. Pagès, “Functional quantization of Gaussian processes,” Journal of Functional Analysis, vol. 196, no. 2, pp. 486–531, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Ledoux and M. Talagrand, Probability in Banach Spaces, vol. 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer, Berlin, Germany, 1991.
  24. H. Heuser, Funktionalanalysis, Mathematische Leitfäden, B. G. Teubner, Stuttgart, Germany, 2nd edition, 1986.
  25. H. Luschgy and G. Pagès, “High-resolution product quantization for Gaussian processes under sup-norm distortion,” Bernoulli, vol. 13, no. 3, pp. 653–671, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. A. DeVore and G. G. Lorentz, Constructive Approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1993.
  27. I. Singer, Bases in Banach Spaces. I, vol. 15 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1970.
  28. K. Dzhaparidze and H. van Zanten, “A series expansion of fractional Brownian motion,” Probability Theory and Related Fields, vol. 130, no. 1, pp. 39–55, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. K. Dzhaparidze and H. van Zanten, “Optimality of an explicit series expansion of the fractional Brownian sheet,” Statistics & Probability Letters, vol. 71, no. 4, pp. 295–301, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. K. Dzhaparidze and H. van Zanten, “Krein's spectral theory and the Paley-Wiener expansion for fractional Brownian motion,” The Annals of Probability, vol. 33, no. 2, pp. 620–644, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. V. Li and Q.-M. Shao, “Gaussian processes: inequalities, small ball probabilities and applications,” in Stochastic Processes: Theory and Methods, vol. 19 of Handbook of Statistics, pp. 533–597, North-Holland, Amsterdam, The Netherlands, 2001. View at Google Scholar · View at Zentralblatt MATH
  32. H. Luschgy and G. Pagès, “Functional quantization of a class of Brownian diffusions: a constructive approach,” Stochastic Processes and Their Applications, vol. 116, no. 2, pp. 310–336, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. Graf, H. Luschgy, and G. Pagès, “Optimal quantizers for Radon random vectors in a Banach space,” Journal of Approximation Theory, vol. 144, no. 1, pp. 27–53, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet