The Annals of Applied Probability

On the distribution of Brownian areas

Mihael Perman and Jon A. Wellner

Full-text: Open access

Abstract

We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with $A^+ = \int_0^1 B^+ (t) dt$ and $A_0^+ = \int_0^1 U^+ (t) dt$, we use excursion theory to show that the Laplace transforms $\Psi^+ (s) = E \exp (-sA^+)$ and $\Psi_0^+ (s) = E \exp (-sA_0^+)$ of $A^+$ of $A_0^+$ satisfy $$\int_0^{\infty} e^{-\lambda s \Psi +} (\sqrt{2} s^{3/2}) ds = \frac{\lambda^{-1/2} Ai(\lambda) + (1/3 - \int_0^{\lambda} Ai(t) dt)}{\sqrt{\lambda} Ai(\lambda) - Ai (\lambda)}.$$ and $$\int_0^{\infty} \frac{e^{-\lambda s}{\sqrt{s}} \Psi_0^+ (\sqrt{2} s^{3/2}) ds = 2 \sqrt{\pi} \frac{Ai(\lambda)}{\sqrt{\lambda} Ai'(\lambda) - Ai(\lambda)}.$$ where Ai is Airy's function. At the same time, our approach via excursion theory unifies previous calculations of this type due to Kac, Groeneboom, Louchard, Shepp and Takács for other Brownian areas. Similarly, we use excursion theory to obtain recursion formulas for the moments of the "positive part" areas. We have not yet succeeded in inverting the double Laplace transforms because of the structure of the function appearing in the denominators, namely, $\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)$.

Article information

Source
Ann. Appl. Probab., Volume 6, Number 4 (1996), 1091-1111.

Dates
First available in Project Euclid: 24 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1035463325

Digital Object Identifier
doi:10.1214/aoap/1035463325

Mathematical Reviews number (MathSciNet)
MR1422979

Zentralblatt MATH identifier
0870.60035

Subjects
Primary: 60G15: Gaussian processes 60G99: None of the above, but in this section
Secondary: 60E05: Distributions: general theory

Keywords
Absolute value Airy functions area asymptotic distribution Brownian bridge Brownian excursion Brownian motion Feynman-Kac inversion moments positive part recursion formulas

Citation

Perman, Mihael; Wellner, Jon A. On the distribution of Brownian areas. Ann. Appl. Probab. 6 (1996), no. 4, 1091--1111. doi:10.1214/aoap/1035463325. https://projecteuclid.org/euclid.aoap/1035463325


Export citation

References

  • ABRAMOWITZ, M. and STEGUN, I. A. 1965. Handbook of Mathematical Functions. Dover, New York. Z.
  • BARLOW, M., PITMAN, J. M. and YOR, M. 1989. Une extension multidimensionnelle de la loi de l'arc sinus. Seminaire de Probabilites XXIII. Lecture Notes in Math. 1372 294 314. ´ ´ Springer, New York. Z.
  • BIANE, PH. and YOR, M. 1987. Valuers principales associees aux temps locaux Browniens. Bull. ´ Sci. Math. 111 23 101. Z.
  • BIANE, PH. and YOR, M. 1988. Sur la loi des temps locaux browniens pris en un temps exponentiel. Seminaire de Probabilites XXII. Lecture Notes in Math. 1321 454 465. ´ ´ Springer, New York. Z.
  • BIRNBAUM, Z. W. and TANG, V. K. T. 1964. Two simple distribution-free tests of goodness of fit. Rev. Internat. Statist. Inst. 32 2 13. Z.
  • BORODIN, A. N. 1984. Distribution of integral functionals of a Brownian motion. J. Soviet Math. 27 3005 3022. Z.
  • CHAPMAN, D. G. 1958. A comparative study of several one-sided goodness-of-fit tests. Ann. Math. Statist. 29 655. Z.
  • CIFARELLI, D. M. 1975. Contributi intorno ad un test per l'omogeneita tra du campioni. G. Z. Econom. Ann. Econ. N.S. 34 233 249. Z.
  • DARLING, D. A. 1983. On the supremum of a certain Gaussian process. Ann. Probab. 11 803 806. Z.
  • DURRETT, R. T. and IGLEHART, D. L. 1977. Functionals of Brownian meander and excursion. Ann. Probab. 5 130 135. Z.
  • Dy NKIN, E. B. 1961. Some limit theorems for sums of independent random variables with infinite expectations. Selected Translations in Mathematical Statistics and Probability, IMS AMS 1 171 189. Z.
  • FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York. Z.
  • GETOOR, R. K. 1979. Excursions of a Markov process. Ann. Probab. 7 244 266. Z.
  • GROENEBOOM, P. 1985. Estimating a monotone density. Proceedings of the Berkeley Conference Z. in Honor of Jerzy Ney man and Jack Kiefer L. M. Le Cam and R. A. Olshen, eds. 2. Univ. California Press, Berkeley. Z.
  • GROENEBOOM, P. 1989. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79 109. Z.
  • ITO, K. and MCKEAN, H. P. 1974. Diffusion Processes and Their Sample Paths, 2nd ed. Springer, Berlin. Z.
  • JOHNSON, B. MCK. and KILLEEN, T. 1983. An explicit for the c.d.f. of the L norm of the 1 Brownian bridge. Ann. Probab. 11 807 808. Z.
  • KAC, M. 1946. On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Amer. Math. Soc. 59 401 414. Z.
  • KAC, M. 1951. On some connections between probability theory and differential and integral equations. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 1 189 215. Univ. California Press, Berkeley. Z.
  • LEVY, P. 1948. Processus Stochastiques et mouvement Brownien. Gauthiers, Paris. ´
  • LOUCHARD, G. 1984a. Kac's formula, Levy's local time and Brownian excursion. J. Appl. ´ Probab. 21 479 499. Z.
  • LOUCHARD, G. 1984b. The Brownian excursion area. Comput. Math. Appl. 10 413 417. ErraZ. tum: A12 1986 375. Z.
  • NASAROW, N. 1931. Ueber die Entwicklung einer beliebigen Funktion nach Laguerreschen Poly nomen. Math. Z. 33 481 487. Z.
  • REVUZ, D. and YOR, M. 1994. Continuous Martingales and Brownian Motion, 2nd ed. Springer, New York. Z.
  • RICE, S. O. 1982. The integral of the absolute value of the pinned Wiener process. Ann. Probab. 10 240 243. Z.
  • RIEDWy L, H. 1967. Goodness of fit. J. Amer. Statist. Assoc. 62 390 398. Z.
  • ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusions, Markov Processes and Martingales. Ito Calculus 2. Wiley, New York. Z.
  • SANSONE, G. 1959. Orthogonal Functions. Interscience, New York. Z.
  • SHEPP, L. A. 1982. On the integral of the absolute value of the pinned Wiener process. Ann.
  • TAKACS, L. 1992a. Random walk processes and their various applications. In Probability ´ Z Theory and Applications. Essay s to the Memory of Jozsef Mogy orodi J. Galambos and ´ ´. I. Katai, eds. 1 32. Kluwer, Dordrecht. ´ Z.
  • TAKACS, L. 1992b. Random walk processes and their application in order statistics. Ann. Appl. ´ Probab. 2 435 459. Z.
  • TAKACS, L. 1993a. On the distribution of the integral of the absolute value of the Brownian ´ motion. Ann. Appl. Probab. 3 186 197. Z.
  • TAKACS, L. 1993b. Personal communication with J. A. Wellner. ´ Z.
  • USPENSKY, J. V. 1927. On the development of arbitrary functions in series of Hermite's and Laguerre's poly nomials. Ann. Math. 28 593 619. Z.
  • WOLFRAM, S. 1991. Mathematica, a Sy stem for Doing Mathematics by Computer, 2nd ed. Addison-Wesley, Redwood City, CA.
  • INSTITUTE FOR MATHEMATICS, physiCS UNIVERSITY OF WASHINGTON AND MECHANICS DEPARTMENT OF STATISTICS DEPARTMENT OF MATHEMATICS BOX 354322
  • UNIVERSITY OF LJUBLJANA SEATTLE, WASHINGTON 98195-4322 JADRANSKA 19 E-MAIL: jaw@stat.washington.edu 61111 LJUBLJANA SLOVENIA E-MAIL: Mihael.Perman@fmf.uni-lj.si