Exact Distribution of Verhulst process

TL;DR

This paper derives an exact density formula for the Verhulst process, a functional of geometric Brownian motion, with applications in biology and stochastic volatility, and explores its properties under measure change.

Contribution

It provides the first exact density formula for the Verhulst process and analyzes its properties, including under Girsanov's measure change.

Findings

Exact density formula for the Verhulst process derived.

Special case density when drift = -1/2 obtained.

Verhulst process remains within its class under Girsanov's measure change.

Abstract

We investigate a Verhulst process, which is the special functional of geometric Brownian motion and has many applications, among others in biology and in stochastic volatility models. We present an exact form of density of a one dimensional distribution of Verhulst process. Simple formula for the density of Verhulst process is obtained in the special case, when the drift of geometric Brownian motion is equal to -1/2. Some special properties of this process are discussed, e.g. it turns out that under Girsanov's change of measure a Verhulst process still remains a Verhulst process but with different parameters.