Abstract
The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Risken, H.: The Fokker-Planck Equation. Springer, Berlin (1989)
Van Kampen: Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam (2007)
Gerstner, W., Kistler, W.: Spiking Neuron Models. Cambridge University Press, Cambridge (2002)
Koch, C.: Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, London (1998)
Berg, H.: Random Walks in Biology. Princeton University Press, Princeton (1993)
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley Series in Probability and Statistics. Wiley, New York (2001)
Siegert, A.J.F.: On the first passage time probability problem. Phys. Rev. 81, 617–623 (1951)
Ricciardi, L.M., Sato, S.: First-passage time density and moments of the Ornstein-Uhlenbeck process. J. Appl. Probab. 25, 43 (1988)
Leblanc, B., Renault, O., Scaillet, O.: A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary. Finance Stoch. 4(1), 109–111 (2000)
Alili, L., Patie, P., Pedersen, J.L.: Representations of the first hitting time density of an Ornstein-Uhlenbeck process. Stoch. Models 21, 967–980 (2005)
Giraudo, M.T., Sacerdote, L., Zucca, C.: A Monte Carlo method for the simulation of first passage times for diffusion processes. Methodol. Comput. Appl. Probab. 3, 215–231 (2001)
Bouleau, N., Lepingle, D.: Numerical Methods for Stochastic Processes. Wiley Series in Probability and Statistics. Wiley, New York (1994)
Gardiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (2004)
Klauder, J.L., Petersen, W.P.: Numerical integration of multiplicative-noise stochastic differential equations. SIAM J. Numer. Anal. 22, 1153–1166 (1985)
Fox, R.F.: Second-order algorithm for the numerical integration of colored-noise problems. Phys. Rev. A 46, 2649–2654 (1991)
Honeycutt, R.L.: Stochastic Runge-Kutta algorithms. I. White noise. Phys. Rev. A 45, 600–603 (1992)
Cecchi, G.A., Magnasco, M.O.: Negative resistance and rectification in Brownian transport. Phys. Rev. Lett. 76, 1968–1971 (1996)
Bao, J., Abe, Y., Zhuo, Y.: An integral algorithm for numerical integration of one-dimensional additive colored noise problems. J. Stat. Phys. 90, 1037–1045 (1997)
Hida, T.: Brownian Motion. Springer, Berlin (1980)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, Berlin (2000)
Pyke, R.: The Haar Function Construction of Brownian Motion Indexed by Sets. Springer, Berlin (1983)
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
Rights and permissions
About this article
Cite this article
Taillefumier, T., Magnasco, M.O. A Haar-like Construction for the Ornstein Uhlenbeck Process. J Stat Phys 132, 397–415 (2008). https://doi.org/10.1007/s10955-008-9545-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9545-8