Abstract
This paper studies the large-t behavior of the boundary generated by the method of images for the first-passage-time problem. We show that this behavior is characterized by certain properties of the Laplace transform of the input measure. Such properties also determine the asymptotic behavior of the first-passage-time density. Most of the paper assumes a positive input measure, which generates a concave boundary. The last section, however, discusses a non-positive measure. We obtain a sufficient condition for the boundary to be convex.
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References
Abramowitz, M., & Stegun, I. (1964). Handbook of mathematical functions. New York: Dover.
Cannon, J. (1984). The one-dimensional heat equation. Cambridge: Cambridge University Press.
Daniels, H. (1974). The maximum size of a closed epidemic. Advances in Applied Probability, 6, 607–621.
Daniels, H. (1982). Sequential tests constructed from images. The Annals of Statistics, 10, 394–400.
Daniels, H. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli, 2, 133–143.
Feller, W. (1966). An introduction to probability theory and its applications (Vol. II). New York: Wiley.
Jaimungal, S., Kreinin, A., & Valov, A. (2009). Integral equations and the first passage time of Brownian motion. Working paper, U. Toronto.
Karlin, S. (1968). Total positivity. Stanford: Stanford University Press.
Lerche, H. (1986). Boundary crossing of Brownian motion. Berlin: Springer.
Lo, V., Roberts, G., & Daniels, H. (2002). Inverse method of images. Bernoulli, 8, 53–80.
Peskir, G. (2002). Limit at zero of the Brownian first-passage density. Probability Theory and Related Fields, 124, 100–111.
Robbins, H., & Siegmund, D. (1970). Boundary crossing probabilities for the Weiner process and sample sums. The Annals of Mathematical Statistics, 41, 1410–1429.
Redner, S. (2001). A guide to first-passage processes. Cambridge: Cambridge University Press.
Song, J., & Zipkin, P. (2012, to appear). Supply streams. Manufacturing and Service Operations Management.
Song, J., & Zipkin, P. (2011). An approximation for the inverse first-passage time problem. Advances in Applied Probability, 43, 264–275.
Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. In Proc. 5th Berkeley symp. math. stat. prob. (Vol. 3, pp. 315–343). Berkeley: University of California Press.
Zipkin, P. (2012). Linear programming and the inverse method of images. Annals of Operations Research. doi:10.1007/s10479-012-1249-4.
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In memory of Cyrus Derman.
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Zipkin, P. On the method of images and the asymptotic behavior of first-passage times. Ann Oper Res 241, 201–224 (2016). https://doi.org/10.1007/s10479-012-1295-y
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DOI: https://doi.org/10.1007/s10479-012-1295-y