Project Materials

Foundation of Stochastic Modeling and Applications

ABSTRACT

This thesis presents an overview of 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 the “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 Wiener integral.

Keywords: Stochastic Processes, Stopping times, Martingales, Galton-Watson branching process, Brownian motion.

TABLE OF CONTENTS

1. Introduction 23
2. Conditional Expectation 25
3. Definitions and Basic Properties 28
4. Maximal Inequalities 33
5. Almost sure convergence of Super or Sub-Martingale and Krickeberg Decomposition 38
6. L1 convergence and Regular Martingales 42
7. Doob’s Decomposition for a sub-martingale 49
Chapter 4. Watson-Galton Stochastic process: Extinction of populations 51

1. Introduction 51
2. Martingale Approach 52
3. Extinction Probability Approach 56
Part 2. Continuous Stochastic Modeling 65
Chapter 5. Stopping Time and Measurable Stochastic Processes 67
1. Stopped Stochastic processes in the continuous case 67
Chapter 6. Introduction to the Brownian Motion 73
1. Kolmogorov Construction of the Brownian Motion 73
2. Characterizations and Transformations of the Brownian Motion 76
3. Transformations 78
4. Standard Brownian Motion 80

CONTENTS iii
5. Elements of random Analysis using the standard Brownian motion 92
Chapter 7. Poisson Stochastic Processes 111
1. Description by exponential inter-arrival 111
2. Counting function 115
3. Approach of the Kolmogorov Existence Theorem 121
4. More properties for the Standard Poisson Process 124
5. Kolmogorov equations 138

Part 3. Stochastic Integration 147
Chapter 8. Itˆo Integration or Stochastic Calculus 149
1. Regularity of paths of stochastic processes 150
2. Definition and justification of the Itˆo Stochastic integrals 153
3. The Itˆo Integral 166
4. Computations 167

Conclusions and Perspectives 175
5. Achievements 175
6. Perspectives 176
Bibliography 177

CHAPTER ONE

General Introduction

1. The context

The present dissertation should be placed in the project to build within the African University of Sciences and Technologies a research team in Stochastic’s and Statistics.

For a significant number of years, the course Measure Theory and Integration (MTI) is taught. In the two precedent Master classes, the course (MTI) has been extensively developed. The time allocated to this course allows now to cover the contents of the main reference of the course which is the exposition of Lo (2018).

That content exposed in seven hundred pages is intended to allow the reader to train himself on the knowledge broken into exercises.

This full course of (MTI) should be the basis of two teams of research in AUST:
(A) a team of research in Abstract integration and in Set-valued Integrations.

2 1. GENERAL INTRODUCTION

(B) A team on Stochastic’s and applications in Finance, Biology, Genetics, Population, etc.

The basis in Probability theory which is beneath (B) will lead to a branch of research in :

(C) Statistical Methods and Applied Statistics.

In setting up the described process, in its Probability theory component, the first step consisted in the development of the course of Foundation of Probability Theory (MFPT) (Lo (2018)).

This book was exposed in 2019 as a PhD course in AUST.

The aim of this dissertation is to gather the mathematical tools for stochastic modeling, or at least to gather a great deal of them in a consistent text based on the books of (MTI) and (MFPT).

So, the dissertation will open the doors of first thesis in Stochastic’s in AUST or will serve the future candidates for theses in Stochastic’s In AUST.

2. Stochastic Modeling

In real, many phenomena are described by sequence of random variables or family of random variables. Those described by a sequence require discrete stochastic modeling while those described

2. STOCHASTIC MODELING 3

by an arbitrary family requires continuous stochastic modeling.

For example :

(a) In gambling, the surplus of a gambler at a discrete time n is a random variable Xn. Here one may be interested in the possibility of the gambler losing all of his money and to get ruined.

(b) Let us assume that some population begins with a patriarch which reproduces a random number offspring at time n = 1. At any time n+1, each of the offspring reproduced at time n gives a random number of offspring. So the total number of new members at time n is a random number Xn.

A natural question is: is there any possibility that the population comes to extinction, that is no offspring are made at some time N. We might also want to have an estimation of the number of offspring for large values of n, whether Xn becomes stable or increases to infinity (case of China in the past) or decreases to zero (actual situation in some European countries).

In these two cases, we face discrete stochastic modeling.

(c) Let us suppose that an insurance company has a surplus St at time t. It continues collecting premiums from clients with Pt the total of premium collected at time t, the return of its investments of the premium with Ct the total investments returns at time t and paying the claims to clients with Lt the total amount

4 1. GENERAL INTRODUCTION

of losses payed to clients. The surplus of the company at time t is
St = u + Pt + Ct 􀀀 Lt;

where u is the initial surplus at tome t = 0 or capital. The worse event the company wants to avoid is the ruin situation at time a t0, which is the first time where St 0.

Dealing with Situation (c) is done through continuous time stochastic modeling.

In this dissertation, we will provide interesting parts of the theory beneath such stochastic modeling.

3. Scope of the dissertation

We divide the dissertation into three parts.

The first part deals with discrete stochastic modeling. We will introduce two very important notions, that is, the notion of stopping times and theory of martingales.

As a first example, we study the extinction question of a sequence of a population, as described in Situation (b) above in specific conditions.

3. SCOPE OF THE DISSERTATION 5

The second part deals with continuous stochastic modeling. Here again, We will introduce to continuous versions for stopping times and most importantly, we are going to complete this section with an introduction to Brownian Motion and present a thorough study of it.

The third part is an opening to Stochastic Integration and Stochastic Differential equations.

Generally, the contents I summarized here can be found in the most important books of the discipline. However, I particularly used Lo`eve (1997), Chung (1974), Neveu (1965) and Lo (2018) for the fundamental modern probability theory, Neveu (1975) for discrete martingale, Billingsley (1995),

Taylor and Karlin (1987) for the introduction to stochastic processes and Kuo (2000) for the stochastic calculus. Gathering all this materials and using them in a coherent way was possible in the frame of the series on probability and statistics in which Professor Lo introduces to the most inner secret of those disciplines in a series of books (Lo (2018), Lo (2018), Lo (2019), etc.) I am grateful to be able to benefit from that frame that helped me to reach so many things in a few months.

I am aware that reading and mastering the the key elements of Stochastic’s and trying to realize the described content is a very difficult and heavy challenge.