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The moments of the area under reflected Brownian bridge conditional on its local time at zero. (English) Zbl 0965.60081

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.

MSC:

60J65 Brownian motion
60J60 Diffusion processes
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