1 The Definition of a Stochastic Process

1 The Definition of a Stochastic ProcessSuppose that ( ,F,P) is a probability space, and thatX: Ris a random that this means that is a space,Fis a -algebra of subsets of ,Pis a countablyadditive, non-negative measure on ( ,F) with total massP( ) = 1, andXis a measurablefunction, ,X 1(B) ={ :X( ) B} Ffor every Borel setB B(R).Astochastic processis simply a collection of random variables indexed bytime. It willbe useful to consider separately the cases of discrete time and continuous time. We willeven have occasion to consider indexing the random variables bynegative time. That is,adiscrete time Stochastic processX={Xn, n= 0,1,2, ..}is a countable collection ofrandom variables indexed by the non-negative integers, and acontinuous time stochasticprocessX={Xt,0 t < }is an uncountable collection of random variables indexed bythe non-negative real general, we may consider any indexing setI Rhaving infinite cardinality, so thatcallingX={X , I}a Stochastic Process simply means thatX is a rando

1 The Definition of a Stochastic Process Suppose that (Ω,F,P) is a probability space, and that X : Ω → R is a random variable. Recall that this means that Ω is a space, F is a σ-algebra of subsets of Ω, P is a countably

Tags:

  Process, Stochastic, Stochastic process

Information

Related search queries