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.

arXiv comment17 pages, 4 figures

References in corpus (8)

• Dynamical foundations of nonextensive statistical mechanics
• Power Laws are Logarithmic Boltzmann Laws
• Generalised geometric Brownian motion: Theory and applications to option pricing
• A solvable model of fracture with power-law distribution of fragment sizes
• Theory and data analysis of player and team ball possession time in football
• Analysis of a stochastic model for bacterial growth and the lognormality of the cell-size distribution
• Deformation of power law in the double Pareto distribution using uniformly distributed observation time
• Analysis of size distributions of fictional characters: from Pokémon to Godzilla