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A characterization of the first hitting time of double integral processes to curved boundaries

Part of: Special processes Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Jonathan Touboul  and
Olivier Faugeras
Jonathan Touboul*
Affiliation:
Odyssée Laboratory, INRIA/ENS/ENPC
Olivier Faugeras*
Affiliation:
Odyssée Laboratory, INRIA/ENS/ENPC
*
∗ Postal address: INRIA, Sophia-Antipolis, 2004 route des Lucioles, BP 93 06902, Sophia-Antipolis Cedex, France.
∗ Postal address: INRIA, Sophia-Antipolis, 2004 route des Lucioles, BP 93 06902, Sophia-Antipolis Cedex, France.
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Abstract

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The problem of finding the probability distribution of the first hitting time of a double integral process (DIP) such as the integrated Wiener process (IWP) has been an important and difficult endeavor in stochastic calculus. It has applications in many fields of physics (first exit time of a particle in a noisy force field) or in biology and neuroscience (spike time distribution of an integrate-and-fire neuron with exponentially decaying synaptic current). The only results available are an approximation of the stationary mean crossing time and the distribution of the first hitting time of the IWP to a constant boundary. We generalize these results and find an analytical formula for the first hitting time of the IWP to a continuous piecewise-cubic boundary. We use this formula to approximate the law of the first hitting time of a general DIP to a smooth curved boundary, and we provide an estimation of the convergence of this method. The accuracy of the approximation is computed in the general case for the IWP and the effective calculation of the crossing probability can be carried out through a Monte Carlo method.

Keywords

First hitting timefirst passage timecurved boundaryintegrated Wiener processdouble integral processnumerical computation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G15: Gaussian processes 60K40: Other physical applications of random processes 65C05: Monte Carlo methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 501 - 528
Copyright
Copyright © Applied Probability Trust 2008 

References

Anderssen, R. S., Hood, F. R. D. and Weiss, R. (1973). On the numerical solution of Brownian motion processes. J. Appl. Prob. 10, 409–418.Google Scholar
Borovkov, K. and Novikov, A. (2005). Explicit bounds for approximation rates for boundary crossing probabilities for the Wiener process. J. Appl. Prob. 42, 82–92.CrossRefGoogle Scholar
Buonocore, A., Nobile, A. G. and Ricciardi, L. M. (1987). A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Prob. 19, 784–800.CrossRefGoogle Scholar
Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.Google Scholar
Durbin, J. (1985). The first-passage-density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122. (Correction: 25 (1988), 840.)Google Scholar
Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
Favella, L., Reineri, M., Ricciardi, L. and Sacerdote, L. (1982). First passage time problems and related computational methods. Cybernetics Systems 13, 95–128.CrossRefGoogle Scholar
Gerstner, W. and Kistler, W. (2002). Spiking Neuron Models. Cambridge University Press.Google Scholar
Giraudo, M., Sacerdote, L. and Zucca, C. (2001). A Monte Carlo method for the simulation of first passage time diffusion processes. Method. Comput. Appl. Prob. 3, 215–231.Google Scholar
Goldman, M. (1971). On the first passage of the integrated Wiener process. Ann. Mat. Statist. 42, 2150–2155.Google Scholar
Groeneboom, P. (1989). Brownian motion with parabolic drift and Airy functions. Prob. Theory Relat. Fields 81, 79–109.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (1987). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Kloeden, P. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
Lachal, A. (1991). Sur le premier instant de passage de l'intégrale du mouvement brownien. Ann. Inst. H. Poincaré Prob. Statist. 27, 385–405.Google Scholar
Lachal, A. (1996). Quelques martingales associées à l'intégrale du processus d'Ornstein–Uhlenbeck. Application à l'étude des premiers instants d'atteinte. Stoch. Stoch. Rep. 58, 285–302.Google Scholar
Lachal, A. (1997). Les temps de passages successifs de l'intégrale du mouvement brownien. Ann. Inst. H. Poincaré Prob. Statist. 33, 1–36.Google Scholar
Lefebvre, M. (1989). First-passage densities of a two-dimensional process. SIAM J. Appl. Math. 49, 1514–1523.Google Scholar
McKean, H. P. (1963). A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2, 227–235.Google Scholar
Nardo, E. D., Nobile, A. G., Pirozzi, E. and Ricciardi, L. M. (2001). A computational approach to first passage time problems for Gauss–Markov processes. Adv. Appl. Prob. 33, 453–482.CrossRefGoogle Scholar
Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution. J. Appl. Prob. 22, 611–618.Google Scholar
Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985). Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Prob. 22, 360–369.CrossRefGoogle Scholar
Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.CrossRefGoogle Scholar
Patie, P. (2004). On some first passage time problem motivated by financial applications. , ETH Zurich.Google Scholar
Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164.Google Scholar
Ricciardi, L. M. and Sato, S. (1988). First-passage time density and moments of the Ornstein–Uhlenbeck process. J. Appl. Prob. 25, 43–57.CrossRefGoogle Scholar
Ricciardi, L., Sacerdote, L. and Sato, S. (1984). On an integral equation for first passage time probability density function. J. Appl. Prob. 21, 302–314.CrossRefGoogle Scholar
Ripley, B. D. (1987). Stochastic Simulation. John Wiley, New York.CrossRefGoogle Scholar
Stoer, J. and Burlisch, R. (1980). Introduction to Numerical Analysis. Springer, New York.Google Scholar
Touboul, J. and Faugeras, O. (2007). The spikes trains probability distributions: a stochastic calculus approach. J. Physiol. Paris 101, 78–98.Google Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.Google Scholar
Wang, L. and Pötzelberger, K. (2007). Crossing probabilities for diffusion processes with piecewise continuous boundaries. Methodol. Comput. Appl. Prob. 9, 21–40.Google Scholar