Foundation of Stochastic Modeling and Applications
Chapter One
PREAMBLE OF THE STUDY
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.
CHAPTER TWO
Stopping times
Introduction
A stopping time intuitively is a stopping rule. It is used in deciding whether to continue or stop a process on the basis of the present position and past events. It also plays an important role in stochastic calculus as it adds more elegance to the theory of martingales. As an example, let us use the situation of a gambler. Suppose a gambler decides that he is going to stop gambling whenever he has no money left. If Xn denotes the amount of money he has left at the nth round and V denotes the time that he stops gambling, then V is a stopping time since the time he stops depends only on how much he has at the present round and not on future happenings.
Before we see what stopping times are and their propertuies, let us begin by giving a brief explanation on stochastic processes.
Stochastic Processes
Definition 2.1. A stochastic process is defined by the triplet (Ω, A, P),(Xt)t∈T ,(E, B)
where (a) (Ω, A, P) is a probability space.
STOPPING TIMES
(b) The time space T is arbitrary.
(c) The state space (E, B(E)) which is a measurable space.
(d) paths of the stochastic process for any fixed ω ∈ Ω defined as the mapping
T → E
t 7→ Xt(ω) = X(t, ω) such that for any t ∈ T, the mapping
Xt
: (Ω, A) → (E, B)
ω 7→ Xt(ω)
is measurable.
Let T be an ordered set. Then we have the following definition.
Definition 2.2 (Filtration). A filtration is a non-decreasing family (Ft)t∈T of σ-algebras. i.e., for all (s, t) ∈ T
, s ≤ t implies
Fs ⊆ Ft
Definition 2.3. A stochastic process (Xt)t∈T is said to be adapted to a filtration (Ft)t∈T (or simply (Ft)t∈T -adapted if for every t ∈ T, Xt is Ft measurable.
CHAPTER THREE
Martingales
Introduction
The theory of Martingales has had a profound effect on modern probability theory and in statistics. Branches of Probability theory such as stochastic calculus rest on martingale foundations. The theory is extremely applicable and powerful as it has a lot of amazing consequences. Thus, it is necessary for every user of probability to at least know the basics of Martingale theory.
The term martingale originates in gambling theory. In a series of gambles organized in a Casino at times n ∈ 1, 2, .. in some night, if at each game, Xn represents the money he plays anddeposit in the playing table at the game n, We have that as the game goes on, the player accumulates information and facts from the outcomes of the former gambles. At each gamble n, if Fn represents the whole information he has. Before playing, he has to estimate its conditional return which is
E(Xn+1/Fn).
CHAPTER FOUR
Watson-Galton Stochastic process : Extinction of populations
Introduction
Let us consider some organism which reproduces itself by a random and non-negative integer W of offspring with probability law PW given by its pdf with respect to the counting measure. Let Xn denote the number of organisms we have at generation n. Then
X0 = 1 and X1 = W
Now, suppose that each of these X1 organisms produces W1,j organisms, then
X2 =
X
1≤j≤X1
W1,j .
So, at any generation n ≥ 1, the size of population is
Xn+1 =
X
1≤j≤Xn
Wn,j . (REC)
The following hypotheses are assumed:
i The lifespans of the member of one generation is the same.
ii All of the Wi,j’s follow the probability law of PW .
iii The random variables Wi,j are independent.
CHAPTER FIVE
Stopping Time and Measurable Stochastic Processes An introduction to stopping times has been done in chapter 2.
Here we are going to see some properties of stochastic processes and measurability of stopped stochastic process in the continuous case.
1. Stopped Stochastic processes in the continuous case In this section, we take T = R+.
Regularity of Paths of Stochastic Processes We endow T with B(T), the induced borel σ-algebra of R on it.
We have two measurability types for stochastic processes. The global measurability and the progressive measurability.
Global Measurability. The stochastic process (Xt)t∈T is said to be globally measurable if when considered as a function of (t, ω), it is measurable. i.e., if
CHAPTER SIX
Introduction to the Brownian Motion
1. Kolmogorov Construction of the Brownian Motion Family of finite distribution probability laws. Let k ≥ 1 and
let 0 = t0 < t1 < … < tk be k real numbers. Let Y = (Yt1 , Yt2 , …, Ytk
be a random vector with k independent and centered random variables. Assume that for i ∈ {1, 2 . . . , k}, Yti ∼ N (0, ti − ti−1). Consider the transformation
X = (Xt1
, Xt2
, …, Xtk
t = (Yt1
, Yt1 + Yt2
, …, Yt1 + Yt2 + … + Ytk
Then,
Y = (Xt1
, Xt2 − Xt1
, …, Xtk − Xtk−1
Now, for any (y1, y2, . . . , yk)
t ∈ R
k, we have that
CHAPTER SEVEN
Poisson Stochastic Processes
In this chapter, we are going to study the Poisson stochastic from three points of view. This stochastic process is mainly used to described occurrence times of some random events over a continuous time. Here are some general examples :
(a) Arrival times to a desk in some bank.
(b) Occurrence times of failure of machines in a company.
(c) Arrival times of tasks to the central unity of a computer.
(d) etc.
Let us introduce the stochastic process from several points of view.
1. Description by exponential inter-arrival
Let us consider a bank desk which opens at 08H00 for example. Assuming that at that time there is no clients and we denote Z = 0 at that initial time taken as t = 0.
CHAPTER EIGHT
Itˆo Integration or Stochastic Calculus This chapter and the one that follows have a special place in
this book since the functions to integrate have two arguments, one of them being random. In clear, we deal with functions of the form
(0.1) R 3 t → f(t, ω) ∈ R,
where ω is element of a probability space (Ω, A, R). For each fixed ω, Formula 0.1 constitutes a path of f, which is defined as a stochastic process of space time ∅ 6= T ⊂ R if for all t ∈ T,
ω → f(t, ω), is measurable as a mapping from (Ω, A) to (R, B(R)).
Actually, this chapter is destined to readers who have gone through the book of ? of the present series. We begin by recalling basic notions about the paths of stochastic processes.
CHAPTER NINE
Conclusions and Perspectives
Achievements
This dissertation focused on the fundamental tools of stochastic modelling :
(1) The theory of Martingales (discrete) and application.
(2) The foundation of stochastic processes through the Kolmogorov Existence process.
(3) The detailed study of two very important stochastic processes.
(3a) The Brownian process
(3a) The Poison process.
(4) The introduction to Stochastic calculus.
CONCLUSIONS AND PERSPECTIVES
Although its large volume and its deepness, we were be able to achieve it. But the full scope proposed by professeur Lo included :
(5) The Stochastic differential equations
(6) The continuous-times Martingales.
Due to time limitation and certainly the limit of the scope of a master degree, we tried to reach at least the parts 1 to 3.
This leads to consider Parts (5) and (6) as immediate perspectives.
Perspectives
After we complete Part 5 and 6, we would be in a advantageous to go deeply in Stochastics in Finance, through the important book of Shiryaev A. N. (1999) Essentials of Stochastic Finance : Facts, Models, Theory.
Bibliography
- Patrick Billingsley (1995). Probability and Measure. Wiley. Third Edition.
- Kai Lai Chung (1974). A Course in Probability Theory. Academic Press. New-York. (2000). Introduction to Stochastic Integration. Springer.
- Lo, G. S. (2019). Introduction to Stochastic processes. Gaton Berger University.
- Lo, G.S.(2018). Measure Theory and Integration by and for the learner. SPAS Editions. Saint-Louis, Calgary, Abuja.
Doi : http://dx.doi.org/10.16929/sbs/2016.0005, ISBN : 978-2-9559183-5-7. - Lo, G.S.(2018). Mathematical Foundation of Probability Theory. SPAS Books Series. Saint-Louis, Senegal – Calgary, Canada. Doi : http://dx.doi.org/10.16929/sbs/2016.0008. Arxiv : arxiv.org/pdf/1808.01713
- Lo G.S., K. T. A. Ngom M. and Diallo M.(2018). Weak Convergence (IIA) – Functional and Random Aspects of the
Univariate Extreme Value Theory. Arxiv : 1810.01625 - Michel Lo`eve (1997). Probability Theory I. Springer Verlag. Fourth Edition.
