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On the asymptotic behaviour of first-passage-time densities for one-dimensional diffusion processes and varying boundaries

Published online by Cambridge University Press:  01 July 2016

V. Giorno ,
A. G. Nobile  and
L. M. Ricciardi
V. Giorno*
Affiliation:
University of Salerno
A. G. Nobile*
Affiliation:
University of Salerno
L. M. Ricciardi*
Affiliation:
University of Naples
*
∗Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, 84100 Salerno, Italy.
∗Postal address: Dipartimento di Informatica e Applicazioni, University of Salerno, 84100 Salerno, Italy.
∗∗Postal address: Dipartimento di Matematica e Applicazioni, Università di Napoli, Via Mezzocannone 8, 80134 Napoli, Italy.
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Abstract

Making use of the integral equations given in [1], [2] and [3], the asymptotic behaviour of the first-passage time (FPT) p.d.f.'s through certain time-varying boundaries, including periodic boundaries, is determined for a class of one-dimensional diffusion processes with steady-state density. Sufficient conditions are given for the cases both of single and of pairs of asymptotically constant and asymptotically periodic boundaries, under which the FPT densities asymptotically exhibit an exponential behaviour. Explicit expressions are then worked out for the processes that can be obtained from the Ornstein–Uhlenbeck process by spatial transformations. Some new asymptotic results for the FPT density of the Wiener process are finally proved, together with a few miscellaneous results.

Keywords

DIFFUSIONWIENER PROCESSORNSTEIN–UHLENBECK PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 4 , December 1990 , pp. 883 - 914
Copyright
Copyright © Applied Probability Trust 1990 

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References

1. 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, 784–800.Google Scholar
2. Buonocore, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1990) On the two-boundary first-crossing-time problem for diffusion processes. J. Appl. Prob. 27, 102–114.Google Scholar
3. 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
4. Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1989) A symmetry-based constructive approach to probability densities for one dimensional diffusion processes. J. Appl. Prob. 26, 707–721.Google Scholar
5. Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
6. Mandl, P. (1968) Analytical Treatment of One-dimensional Markov Processes . Springer-Verlag, Berlin.Google Scholar
7. Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of Ornstein–Uhlenbeck first passage time densities. J. Appl. Prob. 22, 360–369.Google Scholar
8. Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1985) Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution. J. Appl. Prob. 22, 611–618.Google Scholar
9. Ricciardi, L. M. and Sato, S. (1988) First-passage-time density and moments of the Ornstein–Uhlenbeck process. J. Appl. Prob. 25, 43–57.Google Scholar
10. Sacerdote, L. (1990) Asymptotic behavior of Ornstein–Uhlenbeck first-passage-time density through periodic boundaries. Appl. Stoch. Models Data Anal. 6, 53–57.Google Scholar
11. Sato, S. (1977) Evaluation of first-passage time probability to a square root boundary for the Wiener process. J. Appl. Prob. 14, 850–856.CrossRefGoogle Scholar
12. Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
13. Wong, E. (1984) The construction of a class of stationary Markoff processes. In Stochastic Processes in Mathematics, Physics and Engineering, Proc. Symp. Appl. Math. XVI, Amer. Math. Soc., Providence, RI.Google Scholar