Abstract
The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.
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References
A. Buonocore, A. G. Nobile, and L. M. Ricciardi, “A new integral equation for the evaluation of first-passage-time probability densities,” Advances in Applied Probability vol. 27 pp. 102–114, 1987.
A. Buonocore, A. Di Crescenzo, and E. Di Nardo, “Input-output behavior of a model neuron with alternating drift,” BioSystems vol. 67 pp. 27–34, 2002.
D. Cyranoski, “Swimming against the tide,” Nature vol. 408 pp. 764–766, 2000.
A. Di Crescenzo, “On Brownian motions with alternating drifts,” In R. Trappl (ed.), Cybernetics and Systems 2000, pp. 324–329, Austrian Society for Cybernetic Studies: Vienna, Austria, 2000.
E. Di Nardo, A. G. Nobile, E. Pirozzi, and L. M. Ricciardi, “A computational approach to first-passage-time problems for Gauss-Markov processes,” Advances in Applied Probability vol. 33 pp. 453–482, 2001.
M. Freidlin and H. Pavlopoulos, “On a stochastic model for moisture budget in an Eulerian atmospheric column,” Environmetrics vol. 8 pp. 425–440, 1997.
A. Giorno, A. G. Nobile, and L. M. Ricciardi, “On the evaluation of first-passage-time probability densities via nonsingular equations,” Advances in Applied Probability vol. 21 pp. 20–36, 1989.
M. Greiner, M. Jobmann, and C. Klüppelberg, “Telecommunication traffic, queueing models, and subexponential distributions,” Queueing Systems vol. 33 pp. 125–152, 1999.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland: Amsterdam, 1981.
K. Kitamura, M. Tokunaga, A. Hikikoshi Iwane, and T. Yanagida, “A single myosin head moves along an actin filament with regular steps of 5.3 nanometres,” Nature vol. 397 pp. 129–134, 1999.
J. R. Michael, W. R. Schucany, and R. W. Haas, “Generating random variates using transformations with multiple roots,” The American Statistician vol. 30 pp. 88–90, 1976.
T. Mikosch and A. V. Nagaev, “Large deviations of heavy-tailed sums with applications in insurance,” Extremes vol. 1 pp. 81–110, 1998.
L. M. Ricciardi, A. Di Crescenzo, V. Giorno, and A. G. Nobile, “An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling,” Mathematica Japonica vol. 50 pp. 247–322, 1999.
S. Ross, Introduction to Probability Models, Academic Press: Boston, 4th edition, 1989.
M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press: San Diego, 1994.
B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall: London, 1986.
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AMS 2000 Subject Classification: 60J65, 60G40, 93E30
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Di Crescenzo, A., Di Nardo, E. & Ricciardi, L.M. Simulation of First-Passage Times for Alternating Brownian Motions. Methodol Comput Appl Probab 7, 161–181 (2005). https://doi.org/10.1007/s11009-005-1481-3
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DOI: https://doi.org/10.1007/s11009-005-1481-3