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November 1996 On the distribution of Brownian areas
Mihael Perman, Jon A. Wellner
Ann. Appl. Probab. 6(4): 1091-1111 (November 1996). DOI: 10.1214/aoap/1035463325


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, $\sqrt{\lambda} Ai(\lambda) - Ai'(\lambda)$.


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Mihael Perman. Jon A. Wellner. "On the distribution of Brownian areas." Ann. Appl. Probab. 6 (4) 1091 - 1111, November 1996.


Published: November 1996
First available in Project Euclid: 24 October 2002

zbMATH: 0870.60035
MathSciNet: MR1422979
Digital Object Identifier: 10.1214/aoap/1035463325

Primary: 60G15 , 60G99
Secondary: 60E05

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

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 4 • November 1996
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