An exponential functional of random walks

TL;DR

This paper studies discrete approximations of the exponential functional of Brownian motion using symmetric random walks, revealing their properties, convergence behavior, and providing a new proof of known results.

Contribution

It introduces a novel analysis of discrete exponential functionals, showing their convergence to the continuous case and characterizing their distribution, which differs from the continuous functional.

Findings

Discrete exponential functionals have singular distributions with finite moments.

Discrete functionals converge almost surely to the continuous exponential functional.

Limit distribution matches the reciprocal of a gamma random variable.

Abstract

The aim of this paper is to investigate discrete approximations of the exponential functional $\int_0^{\infty} \exp(B(t) - \nu t) \di t$ of Brownian motion (which plays an important role in Asian options of financial mathematics) by the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular w.r.t. Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge a.s. to the exponential functional of Brownian motion, hence their limit distribution is the same as in the continuous case, namely, the one of the reciprocal of a gamma random variable, so absolutely continuous w.r.t. Lebesgue measure. This way we give a new, elementary proof for an earlier result by Dufresne and Yor as well.