Lesson 3: Basic theory of stochastic
processes
Umberto Triacca
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica
Università dell’Aquila,
[email protected]
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Probability space
We start with some definitions
A probability space is a triple (Ω,A,P), where
(i) Ω is a nonempty set, we call it the sample space.
(ii) A is a σ-algebra of subsets of Ω, i.e. a family of subsets closed
with respect to countable union and complement with respect to
Ω.
(iii) P is a probability measure defined for all members of A. That
is a function P : A → [0,1] such that P(A) ≥ 0 for all A ∈ A,
P(Ω) = 1, P(∪∞i=1Ai ) =
∑∞
i=1 P(Ai ), for all sequences Ai ∈ A
such that Ak ∩ Aj = ∅ for k 6= j .
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Random Variable
A real random variable or real stochastic variable on (Ω,A,P) is a
function x : Ω→ R, such that the inverse image of any interval
(−∞, a] belongs to A, i.e.
x−1((−∞, a]) = {ω ∈ Ω : x(ω) ≤ a} ∈ A for all a ∈ R.
We also say that the function x is A-measurable.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic process
What is a stochastic process?
Let T be a subset of R.
A real stochastic process is a family of random variables
{xt(ω); t ∈ T }, all defined on the same probability space (Ω,A,P)
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
The set T is called index set of the process. If T ⊂ Z, then the
process {xt(ω); t ∈ T } is called a discrete stochastic process. If T
is an interval of R, then {xt(ω); t ∈ T } is called a continuous
stochastic process.
In the sequel we will consider only discrete stochastic processes.
Any single real random variable is a (trivial) stochastic process. In
this case we have {xt(ω); t ∈ T } with T ={t1}
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
When T = Z the stochastic process {xt(ω); t ∈ Z} becomes a
sequence of random variables.
It is important to keep in mind that the sequence
{xt(ω); t ∈ Z}
has to be understood as the function associating the random
variable xt with the integer t. Therefore the processes
x = {xt(ω); t ∈ Z} ,
y = {x−t(ω); t ∈ Z}
z = {xt−3(ω); t ∈ Z}
are different. Although they share the same range, i.e. the same
set of random variables, the functions associating a random
variable with each integer t are different.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes: examples
Let A(ω) be a random variable defined on (Ω,A,P).
Consider the discrete stochastic process
{xt(ω); t ∈ Z}
where xt(ω) = A(ω) ∀t ∈ Z.
A slightly modified example is
xt(ω) = (−1)tA(ω).
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes: examples
Consider the discrete stochastic process
{xt(ω); t ∈ Z}
where the random variables xt1 , xt2 , ..., xts are independent,
identically distributed (iid) for any finite set of indices
{t1, t2, · · · , ts} ⊂ Z with s ∈ Z+.
This process is called iid process.
If the random variables xt have mean 0 and variance σ
2
x then we
will write
xt ∼ iid(0, σ2x)
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes: examples
Other processes are:
{yt(ω); t ∈ Z}, with yt(ω) = a + bt + ut(ω);
{zt(ω); t ∈ Z}, with zt(ω) = tut(ω).
where ut ∼ iid(0, σ2u).
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Let
{xt(ω); t ∈ Z}
be a stochastic process defined on the probability space (Ω,A,P).
For a fixed ω∗ ∈ Ω,
{xt(ω∗); t ∈ Z}
is a sequence of real number called realization or sample function
of the stochastic process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consider an iid process
{xt(ω); t ∈ Z}
where xt(ω) ∼ N (0, 1) for t ∈ Z. The plot of a realization of this
process is presented in Figure 1.
Figure : Figure 1
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
We note that for each choice of ω ∈ Ω a realization of the
stochastic process is determined. For example, if ω1, ω2 ∈ Ω we
have that {xt(ω1); t ∈ Z} and {xt(ω2); t ∈ Z} are two possible
realizations of our stochastic process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consider the discrete stochastic process
{xt(ω); t ∈ N}
where
xt = log(t) + cos (ut(ω))
with ut ∼ iid(0, 1). Figure 2 shows the plot of two possible
realizations of this process.
Figure : An example of 2 realizations corresponding to 2 ω’s.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Just as a random variable assigns a number to each outcome in a
sample space, a stochastic process assigns a sample function
(realization) to each outcome ω ∈ Ω. Each realization is a unique
function of time different from the others.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
The set of all possible realizations of a stochastic process
{{xt(ω); t ∈ Z};ω ∈ Ω}
is called ensemble.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consider a stochastic process {xt(ω); t ∈ Z}. It is important to
point out that all the random variables xt(ω) are defined on the
same probability space (Ω,A,P):
xt : Ω→ R ∀t ∈ Z.
Therefore, for all s ∈ Z+ and t1 ≤ t2 ≤ · · · ≤ ts , the probability
P(a1 ≤ xt1(ω) ≤ b1, a2 ≤ xt2(ω) ≤ b2, . . . , as ≤ xts (ω) ≤ bs)
is well defined and so we can give the following definition.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Definition. Let {t1, t2, · · · , ts} be a finite set of integers, with
s ∈ Z+.The joint distribution function of
(xt1(ω), xt2(ω), ..., xts (ω))
is defined by Ft1,t2,··· ,ts (b1, b2, · · · , bs) = P(xt1(ω) ≤ b1, xt2(ω) ≤
b2, . . . , xts (ω) ≤ bs)
The family{
Ft1,t2,··· ,ts (b1, b2, · · · , bs); s ∈ Z+, {t1, t2, · · · , ts} ⊂ Z
}
is called the finite dimensional distribution of the process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
If we know the finite dimensional distribution of the process, we
are able to answer the questions such as:
1 Which is the probability that the process {xt(ω); t ∈ Z}
passes through [a, b] at time t1?
2 Which is the probability that the process {xt(ω); t ∈ Z}
passes through [a, b] at time t1 and through [c , d ] at time t2?
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
The answers:
1 P({a ≤ xt1(ω) ≤ b}) = Ft1(b)− Ft1(a)
2 P({a ≤ xt1(ω) ≤ b, c ≤ xt2(ω) ≤ d}) =
Ft1,t2(b, d)− Ft1,t2(a, d)− Ft1,t2(b, c) + Ft1,t2(a, c).
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
An important point: Is the knowledge of the finite dimensional
distribution of the process sufficient to answer all question about
the stochastic process are of interest?
Can the probabilistic structure of a stochastic process to be fully
described by the finite dimensional distribution of the process?
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Theorem. For any positive integer s, let {t1, t2, · · · , ts} be any
admissible set of values of t. Then under general conditions the
probabilistic structure of the stochastic process {xt(ω); t ∈ Z} is
completely specified if we are given the joint probability
distribution of (xt1(ω), xt2(ω), , xtn(ω)) for all values of s and for all
choices of {t1, t2, · · · , ts} (Priestly 1981, p.104).
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
We can conclude that a stochastic process is defined completely in
a probabilistic sense if one knows the joint distribution function of
(xt1(ω), xt2(ω), ..., xts (ω))
Ft1,t2,··· ,ts (b1, b2, · · · , bs)
for any positive integer s and for all choices of finite set of random
variables (xt1(ω), xt2(ω), ..., xts (ω)).
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
The stochastic process as model.
If we take the point of view that the observed time series is a finite
part of one realization of a stochastic process {xt(ω); t ∈ Z}, then
the stochastic process can serve as model of the DGP that has
produced the time series.
