Foundation of Stochastic Modeling and Applications

Abstract

This thesis presents an overview on the theory of stopping times, martingales and Brownian motion which are the foundations of stochastic modeling. We started with a detailed study of discrete stopping times and their properties. Next, we reviewed the theory of martingales and saw an application to solving the problem of "extinction of populations". After that, we studied stopping times in the continuous case and finally, we treated extensively the concepts of Brownian motion and the Wienner integral.