## 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

### Citations:

Zbl 0743.62101; Zbl 0416.60086
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