Table of Contents Author Guidelines Submit a Manuscript
International Journal of Stochastic Analysis
Volume 2011, Article ID 247329, 89 pages
http://dx.doi.org/10.1155/2011/247329
Research Article

Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

1Lewis-Sigler Institute, Princeton University, Carl Icahn Laboratory, Princeton, NJ 08544, USA
2Laboratory of Mathematical Physics, The Rockefeller University, New York, NY 10065, USA
3Mathematical Neuroscience Laboratory, Collège de France, CIRB, 11 Place Marcelin Berthelot, CNRS UMR 7241 and INSERM U 1050, Université Pierre et Marie Curie ED, 158 and Memolife PSL, 75005 Paris, France
4INRIA BANG Laboratory, Paris, France

Received 28 April 2011; Accepted 6 October 2011

Academic Editor: Agnès Sulem

Copyright © 2011 Thibaud Taillefumier and Jonathan Touboul. 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. P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Applications of Mathematics, Stochastic Modelling and Applied Probability, Springer, Berlin, Germany, 2nd edition, 2004.
  2. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Classics in Mathematics, Springer, Berlin, Germany, 2006.
  3. T. Lyons, “Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young,” Mathematical Research Letters, vol. 1, no. 4, pp. 451–464, 1994. View at Google Scholar · View at Zentralblatt MATH
  4. R. F. Bass, B. M. Hambly, and T. J. Lyons, “Extending the Wong-Zakai theorem to reversible Markov processes,” Journal of the European Mathematical Society, vol. 4, no. 3, pp. 237–269, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. P. K. Friz and N. B. Victoir, Multidimensional Stochastic Processes as Rough Paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Theory and Applications, Cambridge University Press, Cambridge, UK, 2010. View at Zentralblatt MATH
  6. L. Coutin and N. Victoir, “Enhanced Gaussian processes and applications,” ESAIM. Probability and Statistics, vol. 13, pp. 247–260, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Friz and N. Victoir, “Differential equations driven by Gaussian signals,” Annales de l'Institut Henri Poincaré Probabilités et Statistiques, vol. 46, no. 2, pp. 369–413, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. P. Lévy, Processus Stochastiques et Mouvement Brownien, Suivi d'une note de M. Loève, Gauthier-Villars & Cie, Paris, France, 2nd edition, 1965.
  9. Z. Ciesielski, “Hölder conditions for realizations of Gaussian processes,” Transactions of the American Mathematical Society, vol. 99, pp. 403–413, 1961. View at Google Scholar · View at Zentralblatt MATH
  10. K. Karhunen, “Zur Spektraltheorie Stochastischer Prozesse,” Annales Academiæ Scientiarum Fennicæ. Series A, vol. 1946, no. 34, p. 7, 1946. View at Google Scholar · View at Zentralblatt MATH
  11. M. Loève, Probability Theory, D. Van Nostrand Co., Inc., Princeton, NJ, USA, 3rd edition, 1963.
  12. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Journal of Educational Psychology, vol. 24, no. 7, pp. 498–520, 1933. View at Publisher · View at Google Scholar
  13. J. L. Lumley, “The structure of inhomogeneous turbulent flows,” in Atmospheric Turbulence and Radio Propagation, pp. 166–178, 1967. View at Google Scholar
  14. M. Kac and A. J. F. Siegert, “An explicit representation of a stationary Gaussian process,” Annals of Mathematical Statistics, vol. 18, pp. 438–442, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L^2(R),” Transactions of the American Mathematical Society, vol. 315, no. 1, pp. 69–87, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. Meyer, F. Sellan, and M. S. Taqqu, “Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion,” The Journal of Fourier Analysis and Applications, vol. 5, no. 5, pp. 465–494, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. C. de Boor, A Practical Guide to Splines, vol. 27 of Applied Mathematical Sciences, Springer, New York, NY, USA, Revised edition, 2001.
  18. M. Fukushima, Y. Ōshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, Germany, 1994.
  19. L. D. Pitt, “A Markov property for Gaussian processes with a multidimensional parameter,” Archive for Rational Mechanics and Analysis, vol. 43, pp. 367–391, 1971. View at Google Scholar · View at Zentralblatt MATH
  20. H. P. McKean, Jr., “A winding problem for a resonator driven by a white noise,” Journal of Mathematics of Kyoto University, vol. 2, pp. 227–235, 1963. View at Google Scholar · View at Zentralblatt MATH
  21. M. Goldman, “On the first passage of the integrated Wiener process,” The Annals of Mathematical Statistics, vol. 42, pp. 2150–2155, 1971. View at Google Scholar · View at Zentralblatt MATH
  22. M. Lefebvre and Leonard, “On the first hitting place of the integrated Wiener process,” Advances in Applied Probability, vol. 21, no. 4, pp. 945–948, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. C. de Boor, C. Gout, A. Kunoth, and C. Rabut, “Multivariate approximation: theory and applications. An overview,” Numerical Algorithms, vol. 48, no. 1–3, pp. 1–9, 2008. View at Publisher · View at Google Scholar
  24. G. S. Kimeldorf and G. Wahba, “A correspondence between Bayesian estimation on stochastic processes and smoothing by splines,” Annals of Mathematical Statistics, vol. 41, pp. 495–502, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. G. S. Kimeldorf and G. Wahba, “Spline functions and stochastic processes,” Sankhyā. Series A, vol. 32, pp. 173–180, 1970. View at Google Scholar · View at Zentralblatt MATH
  26. T. Kolsrud, “Gaussian random fields, infinite-dimensional Ornstein-Uhlenbeck processes, and symmetric Markov processes,” Acta Applicandae Mathematicae, vol. 12, no. 3, pp. 237–263, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1992. View at Publisher · View at Google Scholar
  28. T. Kailath, A. Vieira, and M. Morf, “Inverses of Toeplitz operators, innovations, and orthogonal polynomials,” SIAM Review, vol. 20, no. 1, pp. 106–119, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. P. Baldi, “Exact asymptotics for the probability of exit from a domain and applications to simulation,” The Annals of Probability, vol. 23, no. 4, pp. 1644–1670, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. P. Baldi, L. Caramellino, and M. G. Iovino, “Pricing complex barrier options with general features using sharp large deviation estimates,” in Monte Carlo and Quasi-Monte Carlo Methods 1998 (Claremont, CA), pp. 149–162, Springer, Berlin, Germany, 2000. View at Google Scholar · View at Zentralblatt MATH
  31. P. Baldi and L. Caramellino, “Asymptotics of hitting probabilities for general one-dimensional pinned diffusions,” The Annals of Applied Probability, vol. 12, no. 3, pp. 1071–1095, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. E. Gobet, “Weak approximation of killed diffusion using Euler schemes,” Stochastic Processes and Their Applications, vol. 87, no. 2, pp. 167–197, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press, Cambridge, Mass, USA, 1st edition, 2001.
  34. J. Sekine, “Information geometry for symmetric diffusions,” Potential Analysis, vol. 14, no. 1, pp. 1–30, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1991.
  36. V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1998.
  37. M. Barczy and P. Kern, “Representations of multidimensionallinear process bridges,” http://arxiv.org/abs/1011.0067.
  38. J. Schauder, “Bemerkungen zu meiner Arbeit “Zur Theorie stetiger Abbildungen in Funktionalräumen”,” Mathematische Zeitschrift, vol. 26, no. 1, pp. 417–431, 1927. View at Publisher · View at Google Scholar
  39. J. Schauder, “Eine Eigenschaft des Haarschen Orthogonalsystems,” Mathematische Zeitschrift, vol. 28, no. 1, pp. 317–320, 1928. View at Publisher · View at Google Scholar
  40. V. M. Brodskiĭ and M. S. Brodskiĭ, “The abstract triangular representation of bounded linear operators and the multiplicative expansion of their eigenfunctions,” Doklady Akademii Nauk SSSR, vol. 181, pp. 511–514, 1968. View at Google Scholar
  41. M. S. Brodskiĭ, Triangular and Jordan Representations of Linear Operators, vol. 32, American Mathematical Society, Providence, RI, USA, 1971.
  42. I. C. Gohberg and M. G. Kreĭn, Theory and Applications of Volterra operators in Hilbert Space, vol. 24, American Mathematical Society, Providence, RI, USA, 1970.
  43. P. Baldi and L. Caramellino, “Asymptotics of hitting probabilities for general one-dimensional pinned diffusions,” The Annals of Applied Probability, vol. 12, no. 3, pp. 1071–1095, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  44. L. Caramellino and B. Pacchiarotti, “Large deviation estimates of the crossing probability for pinned Gaussian processes,” Advances in Applied Probability, vol. 40, no. 2, pp. 424–453, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  45. C. L. Dolph and M. A. Woodbury, “On the relation between Green's functions and covariances of certain stochastic processes and its application to unbiased linear prediction,” Transactions of the American Mathematical Society, vol. 72, pp. 519–550, 1952. View at Google Scholar · View at Zentralblatt MATH
  46. G. Kimeldorf and G. Wahba, “Some results on Tchebycheffian spline functions,” Journal of Mathematical Analysis and Applications, vol. 33, pp. 82–95, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  47. N. Aronszajn, “Theory of reproducing kernels,” Transactions of the American Mathematical Society, vol. 68, pp. 337–404, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  48. H. H. Kuo, Gaussian Measures in Banach spaces, vol. 463 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.
  49. T. Lyons and Z. Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, UK, 2002. View at Publisher · View at Google Scholar
  50. G. Wahba, Spline Models for Observational Data, vol. 59 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1990.
  51. A. Rakotomamonjy and S. Canu, “Frames, reproducing kernels, regularization and learning,” Journal of Machine Learning Research, vol. 6, pp. 1485–1515, 2005. View at Google Scholar · View at Zentralblatt MATH
  52. T. Hsing and H. Ren, “An RKHS formulation of the inverse regression dimension-reduction problem,” The Annals of Statistics, vol. 37, no. 2, pp. 726–755, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  53. G. Allaire, Numerical Analysis and Optimization, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2007. View at Zentralblatt MATH
  54. M. O. Magnasco and T. Taillefumier, “A Haar-like construction for the Ornstein Uhlenbeck process,” Journal of Statistical Physics, vol. 132, no. 2, pp. 397–415, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  55. J. Touboul and O. Faugeras, “A characterization of the first hitting time of double integral processes to curved boundaries,” Advances in Applied Probability, vol. 40, no. 2, pp. 501–528, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  56. W. Dahmen, B. Han, R.-Q. Jia, and A. Kunoth, “Biorthogonal multiwavelets on the interval: cubic Hermite splines,” Constructive Approximation, vol. 16, no. 2, pp. 221–259, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  57. C. Donati-Martin, R. Ghomrasni, and M. Yor, “On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options,” Revista Matemática Iberoamericana, vol. 17, no. 1, pp. 179–193, 2001. View at Google Scholar · View at Zentralblatt MATH
  58. M. O. Magnasco and T. Taillefumier, “A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries,” Journal of Statistical Physics, vol. 140, no. 6, pp. 1130–1156, 2010, With supplementary material available onlin. View at Publisher · View at Google Scholar
  59. A. Buonocore, A. G. Nobile, and L. M. Ricciardi, “A new integral equation for the evaluation of first-passage-time probability densities,” Advances in Applied Probability, vol. 19, no. 4, pp. 784–800, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  60. E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Computational Neuroscience, MIT Press, Cambridge, Mass, USA, 2007.
  61. B. Ermentrout, “Type I membranes, phase resetting curves, and synchrony,” Neural Computation, vol. 8, no. 5, pp. 979–1001, 1995. View at Google Scholar
  62. G. B. Ermentrout and N. Kopell, “Parabolic bursting in an excitable system coupled with a slow oscillation,” SIAM Journal on Applied Mathematics, vol. 46, no. 2, pp. 233–253, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  63. K. Daoudi, J. Lévy Véhel, and Y. Meyer, “Construction of continuous functions with prescribed local regularity,” Constructive Approximation, vol. 14, no. 3, pp. 349–385, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH