## Advances in Applied Probability

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

### The integral of geometric Brownian motion

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 30 August 2001

**Permanent link to this document**

http://projecteuclid.org/euclid.aap/999187905

**Digital Object Identifier**

doi:10.1239/aap/999187905

**Mathematical Reviews number (MathSciNet)**

MR2002c:60132

**Zentralblatt MATH identifier**

0980.60103

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

Secondary: 91B28

**Keywords**

Brownian motion Bougerol's identity Asian options

#### Citation

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