Geometric Brownian motion with random observation time as generalization of the double Pareto distribution
Abstract
We study the probability distribution of the value of geometric Brownian motion at the stochastic observation time. It is known that the exponentially distributed observation time yields the distribution called the double Pareto distribution, and this study aims to generalize this distribution. First, we provide a calculation formula for the moment of the observed value of geometric Brownian motion using the moment-generating function of the observation time distribution. Next, the probability density of the observed value of geometric Brownian motion is exactly derived under the observation time following the generalized inverse Gaussian distribution. This result includes cases where the observation time follows the gamma, inverse gamma, and inverse Gaussian distributions, and can be regarded as a generalization of the double Pareto distribution.
Keywords
Cite
@article{arxiv.2509.24201,
title = {Geometric Brownian motion with random observation time as generalization of the double Pareto distribution},
author = {Ken Yamamoto and Takashi Bando and Hirokazu Yanagawa and Yorhihiro Yamazaki},
journal= {arXiv preprint arXiv:2509.24201},
year = {2025}
}
Comments
17 pages, 4 figures