The Maximum of Brownian Motion Minus a Parabola

Piet Groeneboom (Delft University of Technology)

Abstract


We derive a simple integral representation for the distribution of the maximum of Brownian motion minus a parabola, which can be used for computing the density and moments of the distribution, both for one-sided and two-sided Brownian motion.

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Pages: 1930-1937

Publication Date: November 17, 2010

DOI: 10.1214/EJP.v15-826

References

  1. M. Abramowitz, and I.E. Stegun. Handbook of Mathematical Functions. National Bureau of Standards Applied Mathematics Series No. 55. U.S. Government Printing Office, Washington. (1964). MR0167642
  2. H.E. Chernoff. Estimation of the mode. Ann. Statist. Math. 16 (1964), 85-99. MR0172382
  3. H.E. Daniels and T.H.R. Skyrme. The maximum of a random walk whose mean path has a maximum. Adv. in Appl. Probab. 17 (1985), 85--99. MR0778595
  4. P. Groeneboom. Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, II. L. M Le Cam and R. A. Olshen, editors, 535 - 555. Wadsworth, Belmont. (1985). MR0822052
  5. P. Groeneboom. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields. 1, (1989), 71--109. MR0981568
  6. P. Groeneboom and J. A. Wellner. Computing Chernoff's distribution. J. Comput. Graph. Statist. 10 (2001), 388--400. MR1939706
  7. S. Janson, G. Louchard and A. Martin-Löf, A. (2010). The maximum of Brownian motion with a parabolic drift. Electronic Journal of Probability, Volume 15, paper 61, 2010
  8. M. Perman and J.A. Wellner. On the distribution of Brownian areas. Ann. Appl. Probab. 6, (1996), 1091--1111. MR1422979
  9. S. Wolfram. Mathematica. (2009). Wolfram Research, Champaign. Math. Review number not available.
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