The Annals of Probability

On the Integral of the Absolute Value of the Pinned Wiener Process

L. A. Shepp

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Abstract

Let $\tilde{W} = \tilde{W}_t, 0 \leq t \leq 1$, be the pinned Wiener process and let $\xi = \int^1_0|\tilde{W}|$. We show that the Laplace transform of $\xi, \phi(s) = Ee^{-\xi s}$ satisfies \begin{equation*}\tag{*}\int^\infty_0 e^{-us}\phi(\sqrt 2 s^{3/2})s^{-1/2} ds = - \sqrt \pi Ai(u)/Ai'(u)\end{equation*} where $Ai$ is Airy's function. Using $(\ast)$, we find a simple recurrence for the moments, $E\xi^n$ (which seem to be difficult to calculate by direct or by other techniques) namely $E\xi^n = e_n \sqrt \pi(36 \sqrt 2)^{-n}/\Gamma \big(\frac{3n + 1}{2}\big)$ where $e_0 = 1, g_k = \Gamma(3k + \frac{1}{2})/\Gamma(k + \frac{1}{2})$ and for $n \geq 1$, $e_n = g_n + \sum^n_{k=1} e_{n-k}\binom{n}{k} \frac{6k + 1}{6k - 1} g_k.$

Article information

Source
Ann. Probab., Volume 10, Number 1 (1982), 234-239.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993926

Digital Object Identifier
doi:10.1214/aop/1176993926

Mathematical Reviews number (MathSciNet)
MR637389

Zentralblatt MATH identifier
0479.60079

JSTOR
links.jstor.org

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

Keywords
Airy moments Kac's method Karhunen-Loeve

Citation

Shepp, L. A. On the Integral of the Absolute Value of the Pinned Wiener Process. Ann. Probab. 10 (1982), no. 1, 234--239. doi:10.1214/aop/1176993926. https://projecteuclid.org/euclid.aop/1176993926


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See also

  • Acknowledgment of Prior Result: L. A. Shepp. Acknowledgment of Priority: On the Integral of the Absolute Value of the Pinned Wiener Process. Ann. Probab., Vol. 19, Iss. 3 (1991), 1397.