Lesson 3: Basic theory of stochastic processes Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it
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=1 A i) = i=1 P(A i), for all sequences A i A such that A k A j = for k j.
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 measurable A.
Stochastic process What is a stochastic process? Let T be a subset of R. A real stochastic process is a family of random variables {x t (ω); t T }, all defined on the same probability space (Ω, A, P)
The set T is called index set of the process. If T Z, then the process {x t (ω); t T } is called a discrete stochastic process. If T is an interval of R, then {x t (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 {x t (ω); t T } with T ={t 1 }
When T = Z the stochastic process {x t (ω); t Z} becomes a sequence of random variables. It is important to keep in mind that the sequence {x t (ω); t Z} has to be understood as the function associating the random variable x t with the integer t. Therefore the processes x = {x t (ω); t Z}, y = {x t (ω); t Z} z = {x t 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.
: examples Let A(ω) be a random variable defined on (Ω, A, P). Consider the discrete stochastic process {x t (ω); t Z} where x t (ω) = A(ω) t Z. A slightly modified example is x t (ω) = ( 1) t A(ω).
: examples Other processes are: {y t (ω); t Z}, with y t (ω) = a + bt + u t (ω); {z t (ω); t Z}, with z t (ω) = tu t (ω). where the random variables u t (ω) are IID.
Let {x t (ω); t Z} be a stochastic process defined on the probability space (Ω, A, P). For a fixed ω Ω, {x t (ω ); t Z} is a sequence of real number called realization or sample function of the stochastic process.
Consider the discrete stochastic process {x t (ω); t N} where x t (ω) N (0, 1) for t = 1, 2... and x t (ω) x s (ω) for t s. The plot of a realization of this process is presented in Figure 1. Figure : Figure 1
We note that for each choice of ω Ω a realization of the stochastic process is determined. For example, if ω 1, ω 2 Ω we have that {x t (ω 1 ); t Z} and {x t (ω 2 ); t Z} are two possible realizations of our stochastic process.
Consider the discrete stochastic process where {x t (ω); t N} x t = log(t) + cos (A(ω)) A(ω) N(0, 1). Figure 2 shows the plot of two possible realizations of this process.
In the following figure we present the plot of five possible realization of a random walk stochastic process Figure :
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.
The set of all possible realizations of a stochastic process is called ensemble. {{x t (ω); t Z}; ω Ω}
Consider a stochastic process {x t (ω); t Z}. It is important to point out that all the random variables x t (ω) are defined on the same probability space (Ω, A, P): x t : Ω R t Z. Therefore, for all s Z + and t 1 t 2 t s, the probability P(a 1 x t1 (ω) b 1, a 2 x t2 (ω) b 2,..., a s x ts (ω) b s ) is well defined and so we can give the following definition.
Definition. Let {t 1, t 2,, t s } be a finite set of integers, with s Z +.The joint distribution function of (x t1 (ω), x t2 (ω),..., x ts (ω)) is defined by F t1,t 2,,t s (b 1, b 2,, b s ) = P(x t1 (ω) b 1, x t2 (ω) b 2,..., x ts (ω) b s ) The family { Ft1,t 2,,t s (b 1, b 2,, b s ); s Z +, {t 1, t 2,, t s } Z } is called the finite dimensional distribution of the process.
Definition. Let {t 1, t 2,, t s } be a finite set of integers, with s Z +. The stochastic process {x t (ω); t Z} is said Gaussian if the joint distribution function of the random vector (x t1 (ω), x t2 (ω),..., x ts (ω)) is normal for any subset of Z, {t 1, t 2,, t s } with s 1. Thus a stochastic process is a Gaussian process if and only if all distribution functions belonging to the finite dimensional distribution of the process are normal. Many real world phenomena are well modeled as Gaussian 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 {x t (ω); t Z} passes through [a, b] at time t 1? 2 Which is the probability that the process {x t (ω); t Z} passes through [a, b] at time t 1 and through [c, d] at time t 2?
The answers: 1 P({a x t1 (ω) b}) = F t1 (b) F t1 (a) 2 P({a x t1 (ω) b, c x t2 (ω) d}) = F t1,t 2 (b, d) F t1,t 2 (a, d) F t1,t 2 (b, c) + F t1,t 2 (a, c).
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?
Theorem. For any positive integer s, let {t 1, t 2,, t s } be any admissible set of values of t. Then under general conditions the probabilistic structure of the stochastic process {x t (ω); t Z} is completely specified if we are given the joint probability distribution of (x t1 (ω), x t2 (ω),, x tn (ω)) for all values of s and for all choices of {t 1, t 2,, t s } (Priestly 1981, p.104).
We can conclude that a stochastic process is defined completely in a probabilistic sense if one knows the joint distribution function of (x t1 (ω), x t2 (ω),..., x ts (ω)) F t1,t 2,,t s (b 1, b 2,, b s ) for any positive integer s and for all choices of finite set of random variables (x t1 (ω), x t2 (ω),..., x ts (ω)).
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 {x t (ω); t Z}, then the stochastic process can serve as model of the DGP that has produced the time series.
DGP SP x 1,..., x T
In particular, since a complete knowledge of a stochastic process requires the knowledge of the finite dimensional distribution of the process, the time series model is given by the family {F t1,t 2,,t s (b 1, b 2,, b s ); s 1, {t 1, t 2,, t s } Z} where the form of the joint distribution functions F t1,t 2,,t s (b 1, b 2,, b s ) is supposed known. It is clear that, in general, this model contains too unknown parameters to be estimated from observed data.
If, for example, we assume that our model is the stochastic process {x t (ω); t Z}, where x t N(µ t, σt 2 ) we have that { b ( ) } 1 v 2 µt F t (b) = exp dv for t = 0 ± 1,... 2πσ 2 t Thus considering only the univariate distributions, we have to estimate a infinite number of parameters {µ t, σ t ; t Z}. σ t
This task is impossible
Consequently, some restrictions have to be made concerning the stochastic process that is adopted as model. In particular, we will consider 1 restrictions on the time-heterogeneity of the process; 2 restrictions on the memory of the process.
The first kind of restrictions enables us to reduce the number of unknown parameters. The second allows us to obtain a consistent estimate of unknown parameters.