Lesson 3: Basic theory of stochastic processes · be a stochastic process de ned on the probability...

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Lesson 3: Basic theory of stochastic processes Umberto Triacca Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit` a dell’Aquila, [email protected] Umberto Triacca Lesson 3: Basic theory of stochastic processes

Transcript of Lesson 3: Basic theory of stochastic processes · be a stochastic process de ned on the probability...

Page 1: Lesson 3: Basic theory of stochastic processes · be a stochastic process de ned on the probability space (;A ... distribution of the process su cient to answer all question ... Basic

Lesson 3: Basic theory of stochasticprocesses

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,

[email protected]

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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 measurable A.

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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

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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

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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

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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 the random variables ut(ω) are IID.

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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Stochastic processes

Consider the discrete stochastic process

xt(ω); t ∈ Nwhere xt(ω) ∼ N (0, 1) for t = 1, 2... and xt(ω)⊥xs(ω) for t 6= s.The plot of a realization of this process is presented in Figure 1.

Figure : Figure 1Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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Stochastic processes

Consider the discrete stochastic process

xt(ω); t ∈ Nwhere

xt = log(t) + cos (A(ω))

A(ω) ∼ N(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

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Stochastic processes

In the following figure we present the plot of five possiblerealization of a random walk stochastic process

Figure :

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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

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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

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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

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Stochastic processes

Definition. Let t1, t2, · · · , ts be a finite set of integers, withs ∈ Z+. The stochastic process xt(ω); t ∈ Z is said Gaussian ifthe joint distribution function of the random vector(xt1(ω), xt2(ω), ..., xts (ω)) is normal for any subset of Z,t1, t2, · · · , ts with s ≥ 1.

Thus a stochastic process is a Gaussian process if and only if alldistribution functions belonging to the finite dimensionaldistribution of the process are normal.

Many real world phenomena are well modeled as Gaussianprocesses.

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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

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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?

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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

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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

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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

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Stochastic processes

&%'$

DGP

?

SP

7

x1, ..., xT

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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

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Stochastic processes

This task is impossible

Umberto Triacca Lesson 3: Basic theory of stochastic processes

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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

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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