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Lesson 3: Basic theory of stochasticprocesses
Umberto Triacca
Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,
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 closedwith respect to countable union and complement with respect toΩ.
(iii) P is a probability measure defined for all members of A. Thatis 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 afunction 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 variablesxt(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 theprocess xt(ω); t ∈ T is called a discrete stochastic process. If Tis an interval of R, then xt(ω); t ∈ T is called a continuousstochastic process.
In the sequel we will consider only discrete stochastic processes.
Any single real random variable is a (trivial) stochastic process. Inthis 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 asequence 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 randomvariable 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 sameset of random variables, the functions associating a randomvariable 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 indicest1, t2, · · · , ts ⊂ Z with s ∈ Z+.
This process is called iid process.
If the random variables xt have mean 0 and variance σ2x then we
will writext ∼ iid(0, σ2
x)
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
Letxt(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 functionof the stochastic process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consider an iid process
xt(ω); t ∈ Zwhere xt(ω) ∼ N (0, 1) for t ∈ Z. The plot of a realization of thisprocess is presented in Figure 1.
Figure : Figure 1Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
We note that for each choice of ω ∈ Ω a realization of thestochastic process is determined. For example, if ω1, ω2 ∈ Ω wehave that xt(ω1); t ∈ Z and xt(ω2); t ∈ Z are two possiblerealizations of our stochastic process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consider the discrete stochastic process
xt(ω); t ∈ Nwhere
xt = log(t) + cos (ut(ω))
with ut ∼ iid(0, 1). Figure 2 shows the plot of two possiblerealizations 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 asample space, a stochastic process assigns a sample function(realization) to each outcome ω ∈ Ω. Each realization is a uniquefunction 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 topoint out that all the random variables xt(ω) are defined on thesame 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, withs ∈ 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, weare able to answer the questions such as:
1 Which is the probability that the process xt(ω); t ∈ Zpasses through [a, b] at time t1?
2 Which is the probability that the process xt(ω); t ∈ Zpasses 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 dimensionaldistribution of the process sufficient to answer all question aboutthe stochastic process are of interest?
Can the probabilistic structure of a stochastic process to be fullydescribed 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 anyadmissible set of values of t. Then under general conditions theprobabilistic structure of the stochastic process xt(ω); t ∈ Z iscompletely specified if we are given the joint probabilitydistribution of (xt1(ω), xt2(ω), , xtn(ω)) for all values of s and for allchoices 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 ina 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 randomvariables (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 finitepart of one realization of a stochastic process xt(ω); t ∈ Z, thenthe stochastic process can serve as model of the DGP that hasproduced the time series.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
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Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
In particular, since a complete knowledge of a stochastic processrequires the knowledge of the finite dimensional distribution of theprocess, the time series model is given by the family
Ft1,t2,··· ,ts (b1, b2, · · · , bs); s ≥ 1, t1, t2, · · · , ts ⊂ Z
where the form of the joint distribution functionsFt1,t2,··· ,ts (b1, b2, · · · , bs) is supposed known. It is clear that , ingeneral, this model contains too unknown parameters to beestimated from observed data.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
If, for example, we assume that our model is the stochastic processxt(ω); t ∈ Z, where xt ∼ N(µt , σ
2t ) we have that
Ft(b) =
∫ b
−∞
1√2πσ2
t
exp
−(v − µtσt
)2dv for t = 0± 1, ...
Thus considering only the univariate distributions, we have toestimate a infinite number of parameters µt , σt ; t ∈ Z.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
This task is impossible
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
Consequently, some restrictions have to be made concerning thestochastic process that is adopted as model. In particular, we willconsider
1 restrictions on the time-heterogeneity of the process;
2 restrictions on the memory of the process.
Umberto Triacca Lesson 3: Basic theory of stochastic processes
Stochastic processes
The first kind of restrictions enables us to reduce the number ofunknown parameters.
The second allows us to obtain a consistent estimate of unknownparameters.
Umberto Triacca Lesson 3: Basic theory of stochastic processes