ScholarlyCommons

Title

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

Author(s)

Larry A. Shepp, University of Pennsylvania

Document Type

Journal Article

Date of this Version

1982

Publication Source

The Annals of Probability

Volume

10

Issue

1

Start Page

234

Last Page

239

DOI

10.1214/aop/1176993926

Abstract

Let W~=W~t,0≤t≤1, be the pinned Wiener process and let ξ = ∫10|W~|. We show that the Laplace transform of ξ,ϕ(s)=Ee^−ξs satisfies

∫∞₀ e^−us ϕ(2√s^3/2)s^−1/2 ds = −√π Ai(u)/Ai′(u)

where Ai is Airy's function. Using (∗), we find a simple recurrence for the moments, Eξⁿ (which seem to be difficult to calculate by direct or by other techniques) namely Eξⁿ = e_n√π(36√2)⁻ⁿ/Γ(3n+1/2) where e₀ = 1,g_k = Γ(3k+1/2)/Γ(k+1/2) and for n ≥ 1,

e_n=g_n+∑ⁿ_k=1e_n−k(n k) (6k+1)/(6k−1) g_k.

Keywords

airy, moments, Kac's method, Karhunen-Loeve

Recommended Citation

Shepp, L. A. (1982). On the Integral of the Absolute Value of the Pinned Wiener Process. The Annals of Probability, 10 (1), 234-239. http://dx.doi.org/10.1214/aop/1176993926