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.
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 submartingale 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 Tranformations of the
Brownian Motion 76
3. Tranformations 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 Stochastics 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.
1
2 1. GENERAL INTRODUCTION
(B) a team on Stochastics 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 Stochastics
in AUST or will serve the future candidates for theses in
Stochastics 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
Stochastics and trying to realize the described content is a
very difficult and heavy challenge. But with the help of the
6 1. GENERAL INTRODUCTION
leaders of AUST, especially the HOD of Pure and Applied Mathematics,
with the full supervision of professor Gane Samb Lo,
we humbly think that we had a firm introduction to stochastic
modeling and we are ready to go further to research
Do you need help? Talk to us right now: (+234) 08060082010, 08107932631 (Call/WhatsApp). Email: [email protected].
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