On distributions associated with the generalized Levy’S stochastic area formula

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A closed-form expression is obtained for the conditional probability distribution of ∫t0R2sds given Rt, where (Rs, s ≧ 0) is a Bessel process of dimension δ > 0 started from 0, in terms of parabolic cylinder functions. This is done by inverting the following Laplace transform also known as the generalized Lévy’s stochastic area formula: . We also examine the joint distribution of (R2t, ∫t0R2sds.