Advances in Applied Probability

The integral of geometric Brownian motion

Daniel Dufresne

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This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

Article information

Adv. in Appl. Probab. Volume 33, Number 1 (2001), 223-241.

First available in Project Euclid: 30 August 2001

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 91B28

Brownian motion Bougerol's identity Asian options


Dufresne, Daniel. The integral of geometric Brownian motion. Adv. in Appl. Probab. 33 (2001), no. 1, 223--241. doi:10.1239/aap/999187905.

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