The moments of the area under reflected Brownian bridge conditional on its local time at zero
Let \(U_T\) denote the real Brownian bridge of duration \(T\), and \(\ell(x)= L^x_T(|U_T|)\) the total local time at \(x\) of its absolute value. A recursion formula for the conditional moments of \(\int^T_0|U_T(s) |ds\) given \(\ell(0)\) is developed. The method is based on the Kac formula relative to the process \((\ell(x), 1-\int^x_0 \ell(u)du)\) recently introduced by Pitman, and on power series expansions. An integral expression for the joint moments of the areas of the positive and negative parts of the Brownian bridge \(U_T\) is deduced.
scientific article; zbMATH DE number 1507624