Date of this Version
The Annals of Probability
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√s3/2)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 = en√π(36√2)−n/Γ(3n+1/2) where e0 = 1,gk = Γ(3k+1/2)/Γ(k+1/2) and for n ≥ 1,
en=gn+∑nk=1en−k(n k) (6k+1)/(6k−1) gk.
airy, moments, Kac's method, Karhunen-Loeve
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
Date Posted: 27 November 2017
This document has been peer reviewed.