Geometric Brownian motion with random observation time as generalization of the double Pareto distribution

TL;DR

This work analyzes the distribution of a geometric Brownian motion value observed at a random time. Replacing the exponential observation time that yields the double Pareto with the generalized inverse Gaussian (GIG) family extends to gamma, inverse gamma, and inverse Gaussian cases, enabling exact analyses beyond the classic double Pareto. A general moment formula links $E[S_T^m]$ to the MGF of the observation time, while Theorem 2 provides closed-form PDFs for S_T when T is GIG, encompassing several important special cases. The results offer a unified, exact framework for subordinated GBM with rich tail behavior, with implications for financial modeling and stochastic process theory.

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.

Figures (4)

• Figure 1: Log-log graphs of $f_{\mathop{\mathrm{Gamma}}\nolimits}(x)$ for $S_0=1$ and different $k$, $\lambda/\sigma^2$, and $\tilde{\mu}/\sigma^2$. (a) Dependence on $k$: $k=0.5$, $0.7$, $1$, and $2$ with $\lambda/\sigma^2=1$ and $\tilde{\mu}/\sigma^2=1$. (b) Enlarged graph of (a) near $x=S_0=1$. The vertical dashed line indicates $x=S_0$. (c) Dependence on $\tilde{\mu}/\sigma^2$: $\tilde{\mu}/\sigma^2=-1$, $0$, $1$, and $2$ with $k=2$ and $\lambda/\sigma^2=1$; (d) Dependence on $\lambda/\sigma^2$: $\lambda/\sigma^2=0.5$, $1$, $2$, and $4$ with $k=2$ and $\tilde{\mu}/\sigma^2=1$.
• Figure 2: Semi-log graphs of the mean $E[S_T]$ given in Eq. \ref{['eq:invgamma_mean']} as a function of $(2|\tilde{\mu}|-\sigma^2)\theta$, with $k=0.5$, $1$, and $2$ and $S_0=1$.
• Figure 3: Log-log graphs of $f_{\mathop{\mathrm{IGamma}}\nolimits}(x)$ for $S_0=1$. (a) Dependence on $k$: $k=1$, $2$, and $4$ with $\theta\sigma^2=1$ and $\tilde{\mu}/\sigma^2=1$; (b) dependence on $\tilde{\mu}/\sigma^2$: $\tilde{\mu}/\sigma^2=-1$, $0$, $1$, and $2$ with $k=2$ and $\theta\sigma^2=1$; (c) dependence on $\theta\sigma^2$: $\theta\sigma^2=0.5$, $1$, $2$, and $4$ with $k=2$ and $\tilde{\mu}/\sigma^2=1$.
• Figure 4: Log-log graphs of $f_{\mathop{\mathrm{IG}}\nolimits}(x)$ for $S_0=1$. (a) Dependence on $\tau\sigma^2$: $\tau\sigma^2=0.5$, $1$, and $2$ with $\omega\sigma^2=1$ and $\tilde{\mu}/\sigma^2=1$; (b) dependence on $\omega\sigma^2$: $\omega\sigma^2=0.2$, $1$, and $5$ with $\tau\sigma^2=1$ and $\tilde{\mu}/\sigma^2=1$. (c) dependence on $\tilde{\mu}/\sigma^2$: $\tilde{\mu}/\sigma^2=-1$, $0$, $1$, and $2$ with $\tau\sigma^2=1$ and $\omega\sigma^2=1$.

Theorems & Definitions (27)

• Proposition 2.1: Moment of lognormal Crow
• proof
• Theorem 2.1: Moment formula for $S_T$
• proof
• Example 2.1
• Example 2.2
• Example 2.3
• Lemma 3.1
• Theorem 3.1
• proof