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A computational approach to first-passage-time problems for Gauss–Markov processes

Published online by Cambridge University Press:  01 July 2016

E. Di Nardo*
Affiliation:
University of Basilicata
A. G. Nobile*
Affiliation:
University of Salerno
E. Pirozzi*
Affiliation:
University of Reggio Calabria
*
Postal address: Dipartimento di Matematica, Università degli Studi della Basilicata, Via N. Sauro 85, 85100 Potenza, Italy.
∗∗ Postal address: Dipartimento di Matematica e Informatica, Università di Salerno, Via S. Allende, 84081 Baronissi (SA), Italy.
∗∗∗ Postal address: Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Università di Reggio Calabria, Via Graziella, 89100 Reggio Calabria, Italy.

Abstract

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2001 

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References

[1] Abrahams, J. (1981). Some comments on conditionally Markov and reciprocal Gaussian processes. IEEE Trans. Commun. 27, 523525.Google Scholar
[2] Abrahams, J. (1986). A survey of recent progress on level-crossing problems for random processes. In Communications and Networks. A Survey of Recent Advances, eds Blake, I. F. and Poor, H. V. Springer, New York, pp. 625.CrossRefGoogle Scholar
[3] Baker, C. T. H. (1978). The Numerical Treatment of Integral Equations. Oxford University Press.Google Scholar
[4] 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, 784800.CrossRefGoogle Scholar
[5] Daniels, H. E. (1969). The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399408.CrossRefGoogle Scholar
[6] Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133143.CrossRefGoogle Scholar
[7] Delves, L. M. and Walsh, J. (1974). Numerical Solution of Integral Equations. Oxford University Press.Google Scholar
[8] Di Crescenzo, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1997). On first-passage-time and transition densities for strongly symmetric diffusion processes. Nagoya Math. J. 145, 143161.CrossRefGoogle Scholar
[9] Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorem. Ann. Math. Statist. 20, 393403.CrossRefGoogle Scholar
[10] 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, 431453.CrossRefGoogle Scholar
[11] Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99122.CrossRefGoogle Scholar
[12] Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291304.CrossRefGoogle Scholar
[13] Ferebee, B. (1983). An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 63, 115.CrossRefGoogle Scholar
[14] Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sato, S. (1989). On the evaluation of first-passage-time probability densities via nonsingular integral equations. Adv. Appl. Prob. 21, 2036.CrossRefGoogle Scholar
[15] Gutiérrez, R., Ricciardi, L. M., Román, P. and Torres, F. (1997). First-passage-time densities for time-non-homogeneous diffusion processes. J. Appl. Prob. 34, 623631.CrossRefGoogle Scholar
[16] Keilson, J. and Ross, H. F. (1975). Passage times distributions for Gaussian Markov (Orns-te-in–Uhlenbeck) statistical processes. In Selected Tables in Mathematical Statistics, Vol. III. American Mathematical Society, Providence, RI, pp. 233327.Google Scholar
[17] Lánský, P. and Smith, C. E. (1989). The effect of a random initial value in neural first-passage-time models. Math. Biosci. 93, 191215.CrossRefGoogle ScholarPubMed
[18] Mehr, C. B. and McFadden, J. A. (1965). Certain properties of Gaussian processes and their first-passage times. J. R. Statist. Soc. B 27, 505522.Google Scholar
[19] Pickands, J. (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 5173.CrossRefGoogle Scholar
[20] Ricciardi, L. M. and Sato, S. (1983). A note on first passage time for Gaussian processes and varying boundaries. IEEE Trans. Inf. Theory 29, 454457.CrossRefGoogle Scholar
[21] Ricciardi, L. M. and Sato, S. (1986). On the evaluation of first-passage-time densities for Gaussian processes. Signal Processing 11, 339357.CrossRefGoogle Scholar
[22] Sacerdote, L. and Tomassetti, F. (1996). On the evaluations and approximations of first-passage-time probabilities. Adv. Appl. Prob. 28, 270284.CrossRefGoogle Scholar