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On certain exponential functionals of the real Bownian motion. (Sur certaines fonctionnelles exponentielles du mouvement brownien réel.) (French. English summary) Zbl 0758.60085

Let \(B\) be a Brownian motion, \(B_ 0=0\), \(Z_ \nu\) be a gamma variable of index \(\nu\). D. Dufresne’s result, \[ \int_ 0^ \infty e^{aB_ s-bs}ds\overset {\text{law}}= 2/(a^ 2 Z2b/a^ 2) \] [Scand. Actuarial J. 1990, No. 1/2, 39-79 (1990; Zbl 0743.62101)] is shown to be a consequence of R. K. Getoor’s theorem, the last exit time at 1 of the Bessel process of dimension \(2(1+\nu)\) starting at 0 has the same law with \(1/2Z_ \nu\) [Ann. Probab. 7, 864-867 (1979; Zbl 0416.60086)]. Furthermore, \[ \left(\int_ 0^ \infty e^{2B_ s- s}ds,\int_ 0^ \infty e^{-2B_ s- s}ds\right)\overset{\text{law}}= \left(T_ +,{1\over T_ +}+{T_ - \over T_ +^ 2}+V\right) \] is also proved, where \(T_ +\), \(T_ -\), \(V\) are independent, \(T_ \pm\) have the same law with the hitting time at 1 of \(B_ .\) and \(Ee^{-\lambda^ 2 V/2}=(1+\lambda)e^{- \lambda}\).

MSC:

60J65 Brownian motion
60J60 Diffusion processes
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