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On some exponential functionals of Brownian motion. (English) Zbl 0765.60084
Author’s abstract: Distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval \([0,T]\) of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time \(T\) is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

MSC:
60J65 Brownian motion
91B28 Finance etc. (MSC2000)
62E15 Exact distribution theory in statistics
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