Abstract
We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.
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References
Buonocore A, Nobile AG, Ricciardi LM (1987) A new integral equation for the evaluation of first-passage-time probability densities. Adv Appl Probab:784–800
Daniels HE (1982) Sequential tests constructed from images. Ann Stat 10:394–400
Daniels HE (1996) Approximating the first crossing-time density for a curved boundary. Bernoulli 2:133–143
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 Probab 8:431–453
Durbin J, Williams D (1992) The first-passage density of the Brownian motion process to a curved boundary. J Appl Probab 29:291–304
Ferebee B (1982) The tangent approximation to one-sided Brownian exit densities. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61:309–326
Ferebee B. (1983) An asymptotic expansion for one-sided Brownian exit densities. Probab Theory Relat Fields 63:1–15
Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus, 2nd edn. Springer, New York
Lehmann A (2002) Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes. Adv Appl Probab 34:869–887
Lerche HR (1986) Boundary crossing of brownian motion. Lecture notes in statistics, vol 40. Springer, Heidelberg
Molini A, Talkner P, Katul GG, Porporato A (2011) First passage time statistics of Brownian motion with purely time dependent drift and diffusion. Physica A: Statistical Mechanics and its Applications 390:1841–1852
Ricciardi LM, Sacerdote L, Sato S (1984) On an integral equation for first-passage-time probability densities. J Appl Probab 21:302–314
Siegmund D (1986) Boundary crossing probabilities and statistical applications. Ann Stat 14:361–404
Strassen V (1967) Almost sure behaviour of sums of independent random variables and martingales. Math Statist Prob 2:315–343
Taillefumier T, Magnasco MO (2010) A fast algorithm for the first-passage times of Gauss-Markov processes with Hölder continuous boundaries. J Stat Phys 140:1130–1156
Wang L, Pötzelberger K (1997) Boundary crossing probability for brownian motion and general boundaries. J Appl Probab 34:54–65
Wang L, Pötzelberger K (2007) Crossing probability for some diffusion processes with piecewise continuous boundaries. Methodol Comput Appl Probab 9:21–40
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**, Z., Wang, L. First Passage Time for Brownian Motion and Piecewise Linear Boundaries. Methodol Comput Appl Probab 19, 237–253 (2017). https://doi.org/10.1007/s11009-015-9475-2
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DOI: https://doi.org/10.1007/s11009-015-9475-2