Abstract
With appropriate regularity assumptions on the increasing concave function x=β(t)<0, the hitting time density p(t) for a transient curve x=β(t) by a 1-dimensional Brownian motion is shown to satisfy \(p(t) \sim \frac{{(1 - r)}}{{\sqrt {2\pi } }}\frac{{\beta (t) - t\beta '(t)}}{{t^{3/2} }}e^{ - (\phi ^2 (t))2} {\text{, as }}t \to \infty \) Here r is the probability of eventually hitting the curve and φ(t)=t −1/2β(t).
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REFERENCES
Anderson, J. M., and Pitt, L. D. (1996). Probabilistic studies on the size of sets of exceptional growth for non-differentiable functions in A*, Preprint.
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation, Cambridge University Press, Cambridge.
Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary, J. Appl. Prob. 22, 99–122.
Durbin, J. (1992). The first passage density of the Brownian motion process to a curved boundary (Appendix by D. Williams), J. Appl. Prob. 29, 291–304.
Erdős, P. (1942). On the law of the iterated logarithm, Ann. Math. 43, 419–436.
Feller, W. (1943). The general form of the so-called law of the iterated logarithm, Trans. Amer. Math. Soc. 54, 373–402.
Gafurov, M. U., and Rotar', V. I. (1984). On the exit of a random walk from a curvilinear boundary, Theory Prob. Appl. 28, 179–184.
Ito, K., and McKean, H. P. (1964). Diffusion Processes, Springer-Verlag, Berlin.
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion, Lecture Notes in Statistics, Vol. 40, Springer-Verlag, New York.
Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. University of California Press, Proc. 5th Berkeley Symp. Math. Stat. Prob. 3, 315–343.
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Anderson, J.M., Pitt, L.D. Large Time Asymptotics for Brownian Hitting Densities of Transient Concave Curves. Journal of Theoretical Probability 10, 921–934 (1997). https://doi.org/10.1023/A:1022662616608
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DOI: https://doi.org/10.1023/A:1022662616608