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

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 $n \geq 1$, $e_n = g_n + \sum^n_{k=1} e_{n-k}\binom{n}{k} \frac{6k + 1}{6k - 1} g_k.$