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AN INTEGRAL EQUATION FOR THE DISTRIBUTION OF THE FIRST EXIT TIME OF A REFLECTED BROWNIAN MOTION

Part of: Numerical approximation and computational geometry Stochastic processes Volterra integral equations

Published online by Cambridge University Press:  04 December 2009

VICTOR DE-LA-PEÑA ,
GERARDO HERNÁNDEZ-DEL-VALLE  and
CARLOS G. PACHECO-GONZÁLEZ
VICTOR DE-LA-PEÑA*
Affiliation:
Statistics Department, Columbia University, 1255 Amsterdam Ave. Mail Code 4403, New York, NY, USA (email: [email protected])
GERARDO HERNÁNDEZ-DEL-VALLE
Affiliation:
Statistics Department, Columbia University, USA (email: [email protected])
CARLOS G. PACHECO-GONZÁLEZ
Affiliation:
Departamento de Matematicas, CINVESTAV-IPN, A. Postal 14-740, Mexico D.F. 07000, Mexico, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Reflected Brownian motion is used in areas such as physiology, electrochemistry and nuclear magnetic resonance. We study the first-passage-time problem of this process which is relevant in applications; specifically, we find a Volterra integral equation for the distribution of the first time that a reflected Brownian motion reaches a nondecreasing barrier. Additionally, we note how a numerical procedure can be used to solve the integral equation.

Keywords

reflected Brownian motionfirst passage timeVolterra integral equation

MSC classification

Secondary: 60G07: General theory of processes 45D05: Volterra integral equations 65D15: Algorithms for functional approximation
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 4 , April 2009 , pp. 445 - 454
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Bañuelos, D., DeBlassie, R. D. and Smits, R., “The first exit time of a planar Brownian motion from the interior of a parabola”, Ann. Probab. 29 (2001) 882–901.CrossRefGoogle Scholar
[2]Bartle, R. G., The elements of real analysis, 2nd edn (John Wiley & Sons, New York, 1976).Google Scholar
[3]Betensky, R. A., “A boundary crossing probability for the Bessel process”, Adv. Appl. Probab. 30 (1998) 807–830.CrossRefGoogle Scholar
[4]Borodin, A. N. and Salminen, P., Handbook of Brownian motion—facts and formulae (Birkhäuser, Basel, 1996).CrossRefGoogle Scholar
[5]Burkholder, D. L., “Exit times of Brownian motion, harmonic majorization and Hardy spaces”, Adv. Math. 26 (1977) 182–205.CrossRefGoogle Scholar
[6]Cheney, W., Analysis for applied mathematics (Springer, New York, 2001).CrossRefGoogle Scholar
[7]Darling, D. A. and Siegert, A. J. F., “The first passage problem for a continuous Markov process”, Ann. Math. Statist. 24 (1953) 624–639.CrossRefGoogle Scholar
[8]DeBlassie, R. D., “Exit times from cones in R n of Brownian motion”, Probab. Theory Related Fields 74 (1987) 1–29.CrossRefGoogle Scholar
[9]DeBlassie, R. D., “The change of a long lifetime for Brownian motion in a horn-shape domain”, Electron. Comm. Probab. 12 (2007) 134–139.CrossRefGoogle Scholar
[10]De-la-Peña, V. and Hernández-del-Valle, G., “First-passage densities of Brownian motion over a non-decreasing, right continuous barrier”. Unpublished technical report, Department of Statistics, Columbia University, 2007.Google Scholar
[11]Doob, J. L., “Heuristic approach to the Kolmogorov–Smirnov theorems”, Ann. Math. Statist. 20 (1949) 393–403.CrossRefGoogle Scholar
[12]Grebenkov, D. S., “NMR survey of the reflected Brownian motion”, Rev. Modern Phys. 79 (2006) 1077.CrossRefGoogle Scholar
[13]Hernández-del-Valle, G., “First passage time densities of Brownian motion and applications to credit risk”. Ph. D. Thesis, Department of Statistics, Columbia University, 2005.Google Scholar
[14]Karatzas, I. and Shreve, S. E., Brownian motion and stochastic calculus (Springer, New York, 1998).CrossRefGoogle Scholar
[15]Karlin, S. and Taylor, H. M., A first course in stochastic processes (Academic Press, New York, 1975).Google Scholar
[16]Klebaner, F., Introduction to stochastic calculus with applications, 2nd edn (Imperial College Press, London, 2005).CrossRefGoogle Scholar
[17]Kress, R., Numerical analysis (Springer, New York, 1998).CrossRefGoogle Scholar
[18]Levitz, P., Grebenkov, D. S., Zinsmeister, M., Kolwankar, K. M. and Sapoval, B., “Brownian flights over a fractal nest and first-passage statistics on irregular surfaces”, Phys. Rev. Lett. 96 (2006) 180601.CrossRefGoogle Scholar
[19]Li, W. V., “The first exit time of Brownian motion from a parabolic domain”, Electron. Comm. Probab. 12 (2003) 134–139.Google Scholar
[20]Li, W. V., “The first exit time of Brownian motion from an unbounded convex domain”, Ann. Probab. 31 (2003) 1078–1096.CrossRefGoogle Scholar
[21]Lifshits, M. and Shi, Z., “The first exit time of Brownian motion from a parabolic domain”, Bernoulli 8 (2002) 745–765.Google Scholar
[22]Peskir, G., “On integral equations arising in the first-passage problem for Brownian motion”, J. Integral Equations Appl. 14 (2001) 397–423.Google Scholar
[23]Revuz, D. and Yor, M., Continuous Martingales and Brownian motion (Springer, Berlin, 1999).CrossRefGoogle Scholar
[24]Ricciardi, L. M., Sacerdote, L. and Sato, S., “On an integral equation for first-passage-time probability densities”, J. Appl. Probab. 21 (1984) 302–314.CrossRefGoogle Scholar