Foundation of Stochastic Modeling and Applications


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 dis- crete 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 stud- ied stopping times in the continuous case and finally, we treated extensively the concepts of Brownian motion and the Wienner in- tegral.


Certification i

Approval iii
Abstract v
Dedication vii
Acknowledgements ix

Chapter 1. General Introduction 1

1. The context 1
2. Stochastic Modeling 2
3. Scope of the dissertation 4
Part 1. Discrete Stochastic Modeling 7

Chapter 2. Stopping times 9

1. Introduction 9
2. Stochastic Processes 9
3. Basic Definitions 12
4. -algebra generated by a stopping time 15
5. Operations on Stopping Times 19
6. Mesurability of a stopped stochastic process 21

Chapter 3. Martingales 23

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
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


1.1 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 al- low 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 team of research in Abstract integration and in Set-valued Integrations.


Neveu J.(1975). Discrete Parameters Martingales. Elsevier.
Neveu J.(1965). Mathematical Foundation of the Calculus of probability. HOLDEN-DAYD series in Probability and Sta- tistics. San Francisco, London, Amsterdam (Translated by Amiel Feinstein from French)
Taylor H.M and Karlin S.(1987). Introduction to Stochastic Modelling. Third Edition. Academic Press.

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