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Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Axel Lehmann
Axel Lehmann*
Affiliation:
Otto-von-Guericke-Universität Magdeburg
*
∗ Postal address: Faculty of Mathematics, Institute for Mathematical Stochastics, Otto-von-Guericke-Universität Magdeburg, PSF 4120, 39016 Magdeburg, Germany. Email address: axel.lehmann@mathematik.uni-magdeburg.de
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Abstract

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.

Keywords

First passage time densitymoving boundariescontinuous Markov processesBrownian motionOrnstein-Uhlenbeck processVolterra integral equation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 869 - 887
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1] Blake, I. F. and Lindsey, W. C. (1973). Level crossing problems for random processes. IEEE Trans. Inf. Theory 19, 295–315.Google Scholar
[2] Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20, 393–403.Google Scholar
[3] Durbin, J. (1971). Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov–Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
[4] Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.Google Scholar
[5] Ferebee, B. (1982). The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 61, 309–326.Google Scholar
[6] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989). On the evaluation of first-passage-time probability densities via non-singular integral equations. Adv. Appl. Prob. 21, 20–36.Google Scholar
[7] Hoffmann-Jörgensen, J., (1994). Probability With a View Toward Statistics. Chapman and Hall, New York.Google Scholar
[8] Lehmann, A. (1992). Erstpassagenprobleme für ausgewählte Markov-prozesse mit stetigen sowie mit treppenförmigen Realisierungen. Dissertation, Fakultät für Mathematik, TU Magdeburg.Google Scholar
[9] Lehmann, A. (1996). Hölder continuity and differentiability of first passage time distributions for continuous Markov processes. Preprint 96-16, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg. Available at http://www.math.uni-magdeburg.de/preprint/.Google Scholar
[10] Lerche, H. R. (1986). Boundary Crossing of Brownian Motion (Lecture Notes Statist. 40). Springer, Berlin.Google Scholar
[11] Park, C. and Schuurmann, F. J. (1976). Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[12] Pauwels, E. J. (1987). Smooth first-passage densities for one-dimensional diffusions. J. Appl. Prob. 24, 370–377.Google Scholar
[13] Pieper, V. and Tiedge, J. (1983). Zuverlässigkeitsmodelle auf der Grundlage stochastischer Modelle von Verschleißprozessen. Math. Operationsforsch. Statist. Ser. Statist. 14, 485–502.Google Scholar
[14] Pieper, V., Dominé, M. and Kurth, P. (1997). Level crossing problems and drift reliability. Math. Meth. Operat. Res. 45, 347–354.Google Scholar
[15] Ricciardi, L. M. (1976). On the transformation of the diffusion process into the Wiener process. J. Math. Anal. Appl. 54, 185–199.Google Scholar
[16] Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.Google Scholar
[17] Schwabik, Š., Tvrdý, M. and Vejvoda, O. (1979). Differential and Integral Equations. Academia, Prague.Google Scholar
[18] Siegmund, D. (1986). Boundary crossing probabilities and statistical applications. Ann. Statist. 14, 361–404.Google Scholar
[19] Strassen, V. (1967). Almost sure behaviour of sums of independent random variables and martingales. In Proc. 5th Berkeley Symp. Math. Statist. Prob., Vol. 11, Contributions to Probability Theory, Part 1, eds Le Cam, L. M. and Neyman, J., University of California Press, Berkeley, pp. 315–343.Google Scholar
[20] Wang, A. T. (1979). Time-dependent functions of Brownian motion that are Markovian. Ann. Prob. 7, 515–525.Google Scholar