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On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Published online by Cambridge University Press:  01 July 2016

Paavo Salminen
Paavo Salminen*
Affiliation:
Åbo Akademi
*
∗Postal address: Åbo Akademi, Mathematical Institute, SF-20500 Åbo 50, Finland.
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Abstract

Let t → h(t) be a smooth function on ℝ+, and B = {Bs; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables Th: = inf {S; Bs = h(s)} and λth: = sup {s ≦ t; Bs = h(s)}, where t> 0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Keywords

AIRY&S FUNCTIONORNSTEIN–UHLENBECK PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 411 - 426
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Benes, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980) Some solvable stochastic control problems. Stochastics 4, 39–84.Google Scholar
[3] Biane, Ph. and Yor, M. (1987) Valeurs principales associées aux temps locaux Browniens. Bull. Sc. math. (2) 111, 23–101.Google Scholar
[4] Breiman, L. (1966) First exit time from a square root boundary. Proc. 5th Berkley Symp. Math. Statist. Prob. 2, 9–16.Google Scholar
[5] Coddington, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.Google Scholar
[6] Csáki, E., Földes, A. and Salminen, P. (1987) On the joint distribution of the maximum and its location for a linear diffusion. Ann. Inst. H. Poincaré. 23, 179–194.Google Scholar
[7] Daniels, H.E. and Skyrme, T. H. R. (1985) The maximum of a random walk whose mean path has a maximum. Adv. Appl. Prob. 17, 85–99.Google Scholar
[8] Durbin, J. (1985) The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.CrossRefGoogle Scholar
[9] Durrett, R. (1984) Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, California.Google Scholar
[10] Ferebee, B. (1983) An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 63, 1–16.Google Scholar
[11] Ferebee, B. (1982) The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 61, 293–302.Google Scholar
[12] Getoor, R. K. (1987) Excursions of a Markov process. Ann. Prob. 7, 1979, 244–266.Google Scholar
[13] Groeneboom, P. (1987) Brownian motion with a parabolic drift and Airy functions. Prob. Theory and Stochastics. To appear.Google Scholar
[14] Ito, K. and Mckean, H. (1965) Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin.Google Scholar
[15] Jennen, O. and Lerche, H. R. (1981) First exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrscheinlichkeitsth. 55, 133–148.Google Scholar
[16] Karatzas, I. and Shreve, S. E. (1984) Trivariate density of Brownian motion, its local and occupational times, with application to stochastic control. Ann. Prob. 12, 819–828.CrossRefGoogle Scholar
[17] Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
[18] Liptser, R.S. and Shiryayev, A. N., (1977) Statistics of Random Processes, Vol. I. Springer-Verlag, New York.Google Scholar
[19] Novikov, A. A., (1977) On estimates and the asymptotic behaviour of nonexit probabilities of a Wiener process to a moving boundary. Mat. Sb. 110, 1979, 539–550; English trans. in Math. USSR Sb. 38, 1981.Google Scholar
[20] Park, C. and Schuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[21] Salminen, P. (1984) On conditional Ornstein-Uhlenbeck processes. Adv. Appl. Prob. 16, 920–922.CrossRefGoogle Scholar
[22] Strassen, V. (1966) Almost sure behaviour of sums of independent random variables and martingales. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2, 315–343.Google Scholar
[23] Uchiyama, K. (1980) Brownian first exit from and sojourn over one sided moving boundary and applications. Z. Wahrscheinlichkeitsth. 54, 75–116.CrossRefGoogle Scholar