Generalizing Geometric Brownian Motion with Bouncing

Abstract

The trajectories of particles moving in a real line and following the Geometrical Brownian motion have been studied. We take processes and give the generalization of the notions, descriptions and models of Geometrical Brownian motion with bouncing. Moreover, we derive the formulas, which enable us to know the time and positions of the meeting for each pair in the considered collections of particles. We provide important results that show the trajectories of the particles at and after the stopping times. Furthermore, we define the super couple, which achieves the highest number of meetings among all the pairs in the collections. Finally, for a spatial case of our model, we generate the joint distribution of the time between the successive meeting of the bouncing Geometric Brownian motion with bouncing and the change between the positions of the consecutive meeting, which will enable us to predict the next times and positions for the meetings in the future.