extbf {Last Time} extbf {Definition of Stochastic Process} extbf {Some Questions of Interest} extbf {Why Continuous Time?} extbf {Two Ways of Thinking about a SP} extbf {Illustration: Sample Paths} extbf {Finite Dimensional Distributions} extbf {Existence of Processes} extbf {Example 3.1.6} extbf {Versions and Modifications} extbf {$sigma $-field generated by a SP}Last Time•Special Cases of CE•Properties of CE•Convergence of CE’s•Tower property•Taking out what is known•Conditional probabilityToday’s lecture: Section 3.1MATH136/STAT219 Lecture 10, October 13, 2008 – p. 1/11Definition of Stochastic Process•A stochastic process is a collection of random variableson some probability space (Ω, F, IP ): {Xt: t ∈ I} for someindex set I•Discrete time SP: I is countable, e.g.{Xn: n = 0, 1, 2, . . . , N} or {Xn: n = 0, 1, 2, . . .}•Continuous time SP: I is uncountable, e.g.{Xt: 0 ≤ t ≤ T } or {Xt: t ≥ 0}MATH136/STAT219 Lecture 10, October 13, 2008 – p. 2/11Some Questions of Interest•What is the distribution of values of the process at any giventimes?•How do future values of the process depend on past valuesor information?•Does the process “converge” in the long run?•Does the process move continuously in time or does it havejumps? Is it nondecreasing?•When is the first time the process reaches some level?•What is the maximum value of the process over some timeinterval?MATH136/STAT219 Lecture 10, October 13, 2008 – p. 3/11Why Continuous Time?•High frequency observations•Limits of discrete time models•Continuous time models often more tractable than discretetime models•In Finance: no arbitrage theory easier in continuous timeMATH136/STAT219 Lecture 10, October 13, 2008 – p. 4/11Two Ways of Thinking about a SP•For fixed t, Xtis a random variable, i.e.Xt(·) : Ω → IR•For fixed ω, the map t 7→ Xt(ω) is called the sample path(aka trajectory) associated with ωMATH136/STAT219 Lecture 10, October 13, 2008 – p. 5/11Illustration: Sample Paths0 0.5 1 1.5 2 2.5 3−2−1.5−1−0.500.511.5tXt(ω)t → Xt(ω1)t → Xt(ω2)MATH136/STAT219 Lecture 10, October 13, 2008 – p. 6/11Finite Dimensional Distributions•For any integer n < ∞ and times t1, t2, . . . , tn∈ I, the jointdistribution of the random vector (Xt1, Xt2, . . . , Xtn) isdenoted by Ft1,t2,...,tnand given byFt1,t2,...,tn(x1, x2, · · · , xn) = IP (Xt1≤ x1, Xt2≤ x2, · · · , Xtn≤ xn)for all x1, x2, . . . , xn∈ IR•The collection of functions{Ft1,t2,...,tn: n < ∞ and t1, . . . , tn∈ I},is called thefinite dimensional distributions (FDD’S) ofthe stochastic process X.MATH136/STAT219 Lecture 10, October 13, 2008 – p. 7/11Existence of Processes•A collection of distribution functions is consistent iflimxk→∞Ft1,...,tn(x1, · · · , xn)= Ft1,...,tk−1,tk+1,...,tn(x1, . . . , xk−1, xk+1, . . . , xn),for all integers n, times t1, . . . , tn∈ I, x1, . . . , xn∈ IR, andany integer 1 ≤ k ≤ n•If X is a SP then its FDD’s are consistent•Given any consistent collection of FDD’s, there exists aprobability space (Ω, F, IP ) and a stochastic process Xdefined on it such that X has the specified FDD’sMATH136/STAT219 Lecture 10, October 13, 2008 – p. 8/11Example 3.1.6•Consider Ω = [0, 1], with its Borel σ-field, and the uniformprobability measure. Let I = [0, 1].•Define processes Yt(ω) = 0 for all ω ∈ Ω, t ∈ I andXt(ω) =(1, t = ω,0, otherwise•Main point: FDD’s are not enough to determine sample pathproperties of the processMATH136/STAT219 Lecture 10, October 13, 2008 – p. 9/11Versions and Modifications•Two SP’s X and Y are versions of one another if they havethe same FDD’s.•Two SP’s X and Y , defined on the same probability space(Ω, F, IP ), aremodifications of one another ifIP (Xt= Yt) = 1 for all t ∈ I•If X and Y are modifications, then they are also versions ofone another•Two versions do not have to be defined on the sameprobability spaceMATH136/STAT219 Lecture 10, October 13, 2008 – p. 10/11σ-field generated by a SP•The σ-field generated by a SP X, denoted FXis thesmallest σ-field containing σ(Xt) for all t ∈ I.•Information contained in the whole process•Two stochastic processes X and Y are independent if theirgenerated σ-fields FXand FYare independentMATH136/STAT219 Lecture 10, October 13, 2008 – p.