# Lesson 3: Basic theory of stochastic Lesson 3: Basic theory of stochastic processes Umberto Triacca

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Lesson 3: Basic theory of stochastic processes

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Università dell’Aquila,

umberto.triacca@univaq.it

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.

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