The joint distributions of some extremums on geometric Brownian motion

This article investigates the distributions of extremums on Geometric Brownian motion. First, some preliminaries concerning the Laplace-Stieltjes function of the first hitting time are presented. Second, the distribution of the maximum surplus before the process first hits the lower barrier a, the distribution of the maximum surplus before the process ultimately leaves the lower barrier a, and the joint distribution of the maximum surplus and the minimum surplus before the process ultimately leaves the lower barrier a are given. Third, the joint distribution of the maximum surplus before the process first hits the lower barrier a, the maximum surplus between the process first hits the lower barrier a and it ultimately leaves the lower barrier a, and the minimum surplus before it ultimately leaves the lower barrier a are obtained. Finally, some numerical examples are presented to illustrate the distributions of extremums.