Foundation of stochastic modeling and applications

Project Abstract

<p> </p><p>This thesis presents an overview on the theory of stopping times,<br>martingales and Brownian motion which are the foundations of<br>stochastic modeling. We started with a detailed study of discrete<br>stopping times and their properties. Next, we reviewed<br>the theory of martingales and saw an application to solving the<br>problem of “extinction of populations”. After that, we studied<br>stopping times in the continuous case and finally, we treated<br>extensively the concepts of Brownian motion and the Wienner integral.<br>KeyWords. Stochastic Processes, Stopping times, Martingales,<br>Galton-Watson branching process, Brownian motion.</p><p>&nbsp;</p> <br><p></p>

Project Overview

<p> General Introduction<br>1. The context<br>The present dissertation should be placed in the project to build<br>within the African University of Sciences and Technologies a<br>research team in Stochastics and Statistics.<br>For a significant number of years, the course Measure Theory<br>and Integration (MTI) is taught. In the two precedent Master<br>classes, the course (MTI) has been extensively developed. The<br>time allocated to this course allows now to cover the contents<br>of the main reference of the course which is the exposition of<br>Lo (2018).<br>That content exposed in seven hundred pages is intended to allow<br>the reader to train himself on the knowledge broken into<br>exercises.<br>This full course of (MTI) should be the basis of two teams of<br>research in AUST:<br>(A) a team of research in Abstract integration and in Set-valued<br>Integrations.<br>1<br>2 1. GENERAL INTRODUCTION<br>(B) a team on Stochastics and applications in Finance, Biology,<br>Genetics, Population, etc.<br>The basis in Probability theory which is beneath (B) will lead<br>to a branch of research in :<br>(C) Statistical Methods and Applied Statistics.<br>In setting up the described process, in its Probability theory<br>component, the first step consisted in the development of<br>the course of Foundation of Probability Theory (MFPT) (Lo (2018)).<br>This book was exposed in 2019 as a PhD course in AUST.<br>The aim of this dissertation is to gather the mathematical tools<br>for stochastic modeling, or at least to gather a great deal of<br>them in a consistent text based on the books of (MTI) and (MFPT).<br>So, the dissertation will open the doors of first thesis in Stochastics<br>in AUST or will serve the future candidates for theses in<br>Stochastics In AUST.<br>2. Stochastic Modeling<br>In real, many phenomena are described by sequence of random variables<br>or family of random variables. Those described by a sequence<br>require discrete stochastic modeling while those described<br>2. STOCHASTIC MODELING 3<br>by an arbitrary family requires continuous stochastic modeling.<br>For example :<br>(a) In gambling, the surplus of a gambler at a discrete time<br>n is a random variable Xn. Here one may be interested in the<br>possibility of the gambler losing all of his money and to get<br>ruined.<br>(b) Let us assume that some population begins with a patriarch<br>which reproduces a random number offspring at time n = 1. At<br>any time n+1, each of the offspring reproduced at time n gives<br>a random number of offspring. So the total number of new members<br>at time n is a random number Xn. A natural question is<br>: is there any possibility that the population comes to extinction,<br>that is no offspring are made at some time N. We might<br>also want to have an estimation of the number of offspring for<br>large values of n, whether Xn becomes stable or increases to<br>infinity (case of China in the past) or decreases to zero (actual<br>situation in some European countries).<br>In these two cases, we face discrete stochastic modeling.<br>(c) Let us suppose that an insurance company has a surplus St<br>at time t. It continues collecting premiums from clients with<br>Pt the total of premium collected at time t, the return of its<br>investments of the premium with Ct the total investments returns<br>at time t and paying the claims to clients with Lt the total amount<br>4 1. GENERAL INTRODUCTION<br>of losses payed to clients. The surplus of the company at time<br>t is<br>St = u + Pt + Ct ô€€€ Lt;<br>where u is the initial surplus at tome t = 0 or capital. The<br>worse event the company wants to avoid is the ruin situation<br>at time a t0, which is the first time where St 0.<br>Dealing with Situation (c) is done through continuous time stochastic<br>modeling.<br>In this dissertation, we will provide interesting parts of the<br>theory beneath such stochastic modeling.<br>3. Scope of the dissertation<br>We divide the dissertation into three parts.<br>./ The first part deals with discrete stochastic modeling. We<br>will introduce two very important notions, that is, the notion<br>of stopping times and theory of martingales.<br>As a first example, we study the extinction question of a sequence<br>of a population, as described in Situation (b) above in<br>specific conditions.<br>3. SCOPE OF THE DISSERTATION 5<br>./ The second part deals with continuous stochastic modeling.<br>Here again, We will introduce to continuous versions for stopping<br>times and most importantly, we are going to complete this<br>section with an introduction to Brownian Motion and present a<br>thorough study of it.<br>./ The third part is an opening to Stochastic Integration and<br>Stochastic Differential equations.<br>Generally, the contents I summarized here can be found in the<br>most important books of the discipline. However, I particularly<br>used Lo`eve (1997), Chung (1974), Neveu (1965) and Lo<br>(2018) for the fundamental modern probability theory, Neveu<br>(1975) for discrete martingale, Billingsley (1995), Taylor and<br>Karlin (1987) for the introduction to stochastic processes and<br>Kuo (2000) for the stochastic calculus. Gathering all this<br>materials and using them in a coherent way was possible in the<br>frame of the series on probability and statistics in which Professor<br>Lo introduces to the most inner secret of those disciplines<br>in a series of books (Lo (2018), Lo (2018), Lo (2019),<br>etc.) I am grateful to be able to benefit from that frame that<br>helped me to reach so many things in a few months.<br>I am aware that reading and mastering the the key elements of<br>Stochastics and trying to realize the described content is a<br>very difficult and heavy challenge. But with the help of the<br>6 1. GENERAL INTRODUCTION<br>leaders of AUST, especially the HOD of Pure and Applied Mathematics,<br>with the full supervision of professor Gane Samb Lo,<br>we humbly think that we had a firm introduction to stochastic<br>modeling and we are ready to go further to research <br></p>