Abstract
Conditions on the boundary and parameters that produce ordering in the first passage time distributions of two different diffusion processes are proved making use of comparison theorems for stochastic differential equations. Three applications of interest in stochastic modeling are presented: a sensitivity analysis for diffusion models characterized by means of first passage times, the comparison of different diffusion models where first passage times represent an important feature and the determination of upper and lower bounds for first passage time distributions.
Similar content being viewed by others
References
F. D. Assaf, M. Shaked, and J. G. Shanthikumar, “1st passage times with PFR densities,” J. Appl. Prob. vol. 22 pp. 185–196, 1985.
C. A. Ball and A. Roma, “A jump diffusion model for European monetary system,” Journal of International Money and Finance vol. 12, pp. 475–492, 1993
A. Di Crescenzo and A. G. Nobile, “Diffusion approximation to a queueing system with time-dependent arrival and service rates,” Queueing Systems vol. 19 pp. 41–62, 1995.
A. Di Crescenzo and L. M. Ricciardi, “Comparing failure times via diffusion models and likelihood ratio ordering,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences vol. E79-A pp. 1429–1432, 1996.
A. Di Crescenzo and L. M. Ricciardi, “Comparing first-passage-times for semi-Markov skip-free processes,” Stat. Prob. Lett. vol. 30 pp. 247–256, 1996.
A. Di Crescenzo and L. M. Ricciardi, “On a discrimination problem for a class of stochastic processes with ordered first-passage times,” Appl. Stoch. Models Bus. Ind. vol. 17 pp. 205–219, 2001.
L. I. Gal'čuk and M. H. A. Davis, “A note on a comparison theorem for equations with different diffusions,” Stochastics vol. 6 pp. 147–149, 1982.
V. Giorno, P. Lánský, A. G. Nobile, and L. M. Ricciardi, “Diffusion approximation and first-passage-time problem for a model neuron. III. A birth-and-death approach,” Biol. Cybern. vol. 58 pp. 387–404, 1988.
V. Giorno, A. G. Nobile, L. M. Ricciardi, and S. Sato, “On the evaluation of first-passage-time probability densities via non-singular integral equations,” Adv. Appl. Prob. vol. 21 pp. 20–36, 1989.
M. T. Giraudo and L. Sacerdote, “An improved technique for the simulation of first passage times for diffusion processes,” Commun. Statist.-Simula. vol. 28 pp. 1135–1163, 1999.
M. T. Giraudo, L. Sacerdote, and C. Zucca, “A MonteCarlo method for the simulation of first passage times of diffusion processes,” Meth. Comp. Appl. Prob. vol. 3 pp. 215–231, 2001
B. Hajek, “Mean stochastic comparison of diffusions,” Z. Wahrscheinlichkeits-theorie verw. Gebiete vol. 68 pp. 315–329, 1985.
Z. Y. Huang, “A comparison theorem for solutions of stochastic differential equations and its applications,” Proc. A.M.S. vol. 91 pp. 611–617, 1984.
N. Ikeda and S. Watanabe, “Stochastic differential equations and diffusion processes,” North Holland Math. Lib. 1989.
S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, San Diego, 1981.
P. Lánský, L. Sacerdote, and F. Tomassetti, “On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity,” Biol. Cybern. vol. 73 pp. 457–465, 1995.
S. Lee and J. Lynch, “Total positivity of Markov chains and the failure rate character of some first passage times,” Adv. Appl. Prob. vol. 29 pp. 713–732, 1997.
H. J. Li and M. Shaked, “On the first passage times for Markov-processes with monotone convex transition kernels,” Stoch. Proc. Appl. vol. 58 pp. 205–216, 1995.
H. J. Li and M. Shaked, “Ageing first-passage times of Markov processes: A matrix approach,” J. Appl. Prob. vol. 34 pp. 1–13, 1997.
S. Nakao, “Comparison theorems for solutions of one-dimensional stochastic differential equations,” Proceedings of the Second Japan-USSR Symposium on Probability Theory. Lecture Notes in Math., vol. 330, pp. 310–315, Springer: Berlin, 1973.
A. G. Nobile, L. M. Ricciardi, and L. Sacerdote, “Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution,” J. Appl. Prob. vol. 22 pp. 611–618, 1985.
G. L. O'Brien, “A new comparison theorem for solutions of stochastic differential equations,” Stochastics vol. 3 pp. 245–249, 1980.
Y. Ouknine, “Comparison et non-confluence des solutions d'équations differentielles stochastiques unidimensionnelles,” Probab. Math. Statist. vol. 11 pp. 37–46, 1990.
L. M. Ricciardi, Diffusion Processes and Related Topics in Biology. Lecture Notes in Biomathematics vol. 14, Springer Verlag: Berlin, 1977.
L. M. Ricciardi and L. Sacerdote, “The Ornstein-Uhlenbeck process as a model for neuronal activity,” Biol. Cybern. vol. 35 pp. 1–9, 1979.
L. M. Ricciardi and S. Sato, Diffusion processes and first-passage-time problems, Lectures in Applied Mathematics and Informatics, L. M. Ricciardi, Ed. Manchester University Press: Manchester, 1990
L. Sacerdote and C. E. Smith, “A qualitative comparison of some diffusion models for neural activity via stochastic ordering,” Biol. Cybern. vol. 83 pp. 543–551, 2000.
L. Sacerdote and C. E. Smith, “New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural model,” Biosystems vol. 58 pp. 59–65, 2000.
S. Sato, “On the moments of the firing interval of the diffusion approximated model neuron,” Math. Biosci. vol. 39 pp. 53–70, 1978.
M. Shaked and J. G. Shanthikumar, “On the 1st passage times of pure jump-processes,” J. Appl. Prob. vol. 20 pp. 427–446, 1988.
M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Inc.: Boston, 1994.
C. E. Smith, “A note on neuronal firing and input variability,” J. Theor. Biol. vol. 154 pp. 271–275, 1992.
A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley Pub. Comp., Inc.: MA, 1965.
T. Yamada, “On a comparison theorem for solutions of stochastic differential equations and its applications,” J. Math. Kyoto Univ. vol. 13-3 pp. 497–512, 1973.
T. Yamada and Y. Ogura, “On the strong comparison theorems for solutions of stochastic differential equations,” Z. Wahrsch. Verw. Gebiete vol. 56 pp. 3–19, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sacerdote, L., Smith, C.E. Almost Sure Comparisons for First Passage Times of Diffusion Processes through Boundaries. Methodology and Computing in Applied Probability 6, 323–341 (2004). https://doi.org/10.1023/B:MCAP.0000026563.27820.ff
Issue Date:
DOI: https://doi.org/10.1023/B:MCAP.0000026563.27820.ff