The Maximum of Brownian Motion with Parabolic Drift

Svante Janson (Uppsala University)
Guy Louchard (Université Libre de Bruxelles)
Anders Martin-Löf (Stockholm University)

Abstract


We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give new series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1893-1929

Publication Date: November 17, 2010

DOI: 10.1214/EJP.v15-830

References

  1. M. Abramowitz, I. A. ; Stegun, eds., Handbook of Mathematical Functions}. Dover, New York, 1972.
  2. Albright, J. R. Integrals of products of Airy functions. J. Phys. A 10 (1977), no. 4, 485--490. MR0443285 (56 #1655)
  3. J. R. Albright ; E. P. Gavathas, Integrals involving Airy functions. Comment on: ``Evaluation of an integral involving Airy functions'' by L. T. Wille and J. Vennik. J. Phys. A vol 19d (1986), no. 13, 2663--2665.
  4. Barbour, Andrew D. A note on the maximum size of a closed epidemic. J. Roy. Statist. Soc. Ser. B 37 (1975), no. 3, 459--460. MR0397907 (53 #1762)
  5. Barbour, A. D. Brownian motion and a sharply curved boundary. Adv. in Appl. Probab. 13 (1981), no. 4, 736--750. MR0632959 (83d:60090)
  6. Daniels, H. E. The maximum size of a closed epidemic. Advances in Appl. Probability 6 (1974), 607--621. MR0373656 (51 #9856)
  7. Daniels, H. E. The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibres. Adv. in Appl. Probab. 21 (1989), no. 2, 315--333. MR0997726 (90k:60069)
  8. Daniels, H. E.; Skyrme, T. H. R. The maximum of a random walk whose mean path has a maximum. Adv. in Appl. Probab. 17 (1985), no. 1, 85--99. MR0778595 (86h:60158)
  9. Groeneboom, P. Estimating a monotone density. Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983), 539--555, Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA, 1985. MR0822052 (87i:62076)
  10. Groeneboom, Piet. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), no. 1, 79--109. MR0981568 (90c:60052)
  11. Groeneboom, Piet. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), no. 1, 79--109. MR0981568 (90c:60052)
  12. P. Groeneboom ; J. A. Wellner . Computing Chernoff's distribution. J. Comput. Graph. Statist. vol 10 (2001), no. 2, 388--400.
  13. Hörmander, Lars. The analysis of linear partial differential operators. I.Distribution theory and Fourier analysis.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 256. Springer-Verlag, Berlin, 1983. ix+391 pp. ISBN: 3-540-12104-8 MR0717035 (85g:35002a)
  14. Janson, Svante. Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Probab. Surv. 4 (2007), 80--145 (electronic). MR2318402 (2008d:60106)
  15. Janson, Svante. Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence. Ann. Comb. 12 (2009), no. 4, 417--447. MR2496126 (Review)
  16. Lachal, Aimé. Sur la distribution de certaines fonctionnelles de l'intégrale du mouvement brownien avec dérives parabolique et cubique.(French) [Distribution of certain functionals of the integral of Brownian motion with parabolic and cubic drift] Comm. Pure Appl. Math. 49 (1996), no. 12, 1299--1338. MR1414588 (97j:60150)
  17. Lefebvre, Mario. First-passage densities of a two-dimensional process. SIAM J. Appl. Math. 49 (1989), no. 5, 1514--1523. MR1015076 (90i:60056)
  18. G. Louchard, Random walks, Gaussian processes and list structures. Theoret. Comput. Sci. 53 (1987), no. 1, 99--124.
  19. Louchard, G.; Kenyon, Claire; Schott, R. Data structures' maxima. SIAM J. Comput. 26 (1997), no. 4, 1006--1042. MR1460714 (98m:68045)
  20. Martin-Löf, Anders. The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. 35 (1998), no. 3, 671--682. MR1659544 (2000e:92046)
  21. Salminen, Paavo. On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Adv. in Appl. Probab. 20 (1988), no. 2, 411--426. MR0938153 (89i:60161)
  22. Smith, Richard L. The asymptotic distribution of the strength of a series-parallel system with equal load-sharing. Ann. Probab. 10 (1982), no. 1, 137--171. MR0637382 (84h:62029)
  23. Steinsaltz, David. Random time changes for sock-sorting and other stochastic process limit theorems. Electron. J. Probab. 4 (1999), no. 14, 25 pp. (electronic). MR1692672 (2000e:60038)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.