Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T06:32:10.476Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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