On the laws of homogeneous functionals of the Brownian bridge (Q2714356)

The authors derive a general method to compute functionals \(\int^S_0f(B_u)du\) of the Brownian motion \(B=(B_u)\) resp. of the Brownian bridge \(b=(b_u)\). The method is ``elementary'' in the following sense: All results follow from the ``basic identity'' NEWLINE\[NEWLINE A_g\overset{\mathcal D} = (\alpha_1+B_{\alpha_1}\cdot \hat T)^{-1} NEWLINE\]NEWLINE where \(A=(A_t)\) is an increasing process with the same space-time-scaling properties as the Brownian motion, \(\alpha_t\) denotes the inverse of \(A\), furthermore, \(g_t:=\sup(u<t:B_u=0)\), and where \(\hat T\) is an \(1/2\)-stable random variable, independent of \(B\). This basic identity (and some of its corollaries) is used in a series of examples to derive results which had been proved in the past by different methods. The method is used to compute the distributions of \(\sup_{0\leq s\leq t}B^2_s\), resp. of \(|N|\cdot \sup_{0\leq s \leq 1}|b_s|\), where \(N\) denotes a standard Gaussian random variable. In Section 4 it is shown that the method also applies to \(\delta\)-dimensional Bessel processes \((\delta<2)\) replacing the Brownian motion.