Lesson 3: Basic Theory Of Stochastic Processes

Lesson 3: Basic theory of stochasticprocessesUmberto TriaccaDipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità dell’Aquila,umberto.triacca@univaq.itUmberto TriaccaLesson 3: Basic theory of stochastic processes

Probability spaceWe start with some definitionsA 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 A) P(Ω) 1, P( i 1 P(Ai ), for all sequences Ai Ai 1 isuch that Ak Aj for k 6 j.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Random VariableA 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processWhat 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThe 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesWhen 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 processesx {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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processes: examplesLet 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 isxt (ω) ( 1)t A(ω).Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processes: examplesOther 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesLet{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 functionof the stochastic process.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesConsider the discrete stochastic process{xt (ω); t N}where 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesWe 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesConsider the discrete stochastic process{xt (ω); t N}wherext log(t) cos (A(ω))A(ω) N(0, 1). Figure 2 shows the plot of two possiblerealizations of this process.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesIn the following figure we present the plot of five possiblerealization of a random walk stochastic processFigure :Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesJust 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThe set of all possible realizations of a stochastic process{{xt (ω); t Z}; ω Ω}is called ensemble.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesConsider 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 probabilityP(a1 xt1 (ω) b1 , a2 xt2 (ω) b2 , . . . , as xts (ω) bs )is well defined and so we can give the following definition.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesDefinition. 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 } Zis called the finite dimensional distribution of the process.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesDefinition. 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesIf we know the finite dimensional distribution of the process, weare able to answer the questions such as:1Which is the probability that the process {xt (ω); t Z}passes through [a, b] at time t1 ?2Which 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThe answers:1P({a xt1 (ω) b}) Ft1 (b) Ft1 (a)2P({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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesAn 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesTheorem. 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesWe 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThe 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processes' SPDGP&%7 ? x1 , ., xT Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesIn 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 TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesIf, for example, we assume that our model is the stochastic process{xt (ω); t Z}, where xt N(µt , σt2 ) we have that( )Z bv µt 21pdv for t 0 1, .exp Ft (b) σt2πσt2 Thus considering only the univariate distributions, we have toestimate a infinite number of parameters {µt , σt ; t Z}.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThis task is impossibleUmberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesConsequently, some restrictions have to be made concerning thestochastic process that is adopted as model. In particular, we willconsider1restrictions on the time-heterogeneity of the process;2restrictions on the memory of the process.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Stochastic processesThe first kind of restrictions enables us to reduce the number ofunknown parameters.The second allows us to obtain a consistent estimate of unknownparameters.Umberto TriaccaLesson 3: Basic theory of stochastic processes

Lesson 3: Basic theory of stochastic processes Umberto Triacca . Many real world phenomena are well modeled as Gaussian processes. Umberto Triacca Lesson 3: Basic theory of stochastic processes . For any positive integer s, let ft 1;t 2; ;t sgbe any admissi