extbf {Last Time} extbf {Happy Halloween!} extbf {Happy Halloween!} extbf {Continuous Time Filtration & Adapted Process} extbf {Continuous Time Martingale} extbf {Basic Properties} extbf {Zero-Mean, Independent Increment Processes} extbf {Functions of Sub-Martingales} extbf {Right-Continuous Filtrations} extbf {Right-Continuous Filtrations (cont.)}Last Time•Discrete time sub- and super-martingalesToday’s lecture: Sections 4.2MATH136/STAT219 Lecture 16, October 31, 2008 – p. 1/10Happy Halloween!What do you get when you divide thecircumference of a jack-o-lantern by itsdiameter?MATH136/STAT219 Lecture 16, October 31, 2008 – p. 2/10Happy Halloween!What do you get when you divide thecircumference of a jack-o-lantern by itsdiameter?Pumpkin πMATH136/STAT219 Lecture 16, October 31, 2008 – p. 3/10Continuous Time Filtration & Adapted Process•A continuous time filtration is a nondecreasing family{Ft, t ≥ 0} of sub-σ-fields of F, i.e. each Ftis a σ-field andFs⊆ Ftfor all 0 ≤ s ≤ t•A continuous time process {Xt, t ≥ 0} is adapted tofiltration {Ft, t ≥ 0} if for all t ≥ 0 the random variable Xtismeasurable with respect to the σ-field Ft•The canonical filtration of a continuous time SP{Xt, t ≥ 0} is the filtration {FXt, t ≥ 0}, whereFXt= σ(Xu: 0 ≤ u ≤ t)•The canonical filtration is the smallest filtration to which{Xt} is adaptedMATH136/STAT219 Lecture 16, October 31, 2008 – p. 4/10Continuous Time Martingale•A continuous time martingale is a pair {(Xt, Ft), t ≥ 0}which satisfies:◦{Ft} is a filtration◦{Xt} is adapted to {Ft}◦IE|Xt| < ∞ for all t ≥ 0◦IE(Xt|Fs) = Xsfor all 0 ≤ s ≤ t (1)•The pair {(Xt, Ft), t ≥ 0} is a continuous timesub-martingale if (1) above is replaced byIE(Xt|Fs) ≥ Xsfor all 0 ≤ s ≤ t•The pair {(Xt, Ft), t ≥ 0} is a continuous timesuper-martingale if (1) above is replaced byIE(Xt|Fs) ≤ Xsfor all 0 ≤ s ≤ tMATH136/STAT219 Lecture 16, October 31, 2008 – p. 5/10Basic Properties•For martingales:IE(Xt) = IE(X0) for all t ≥ 0•For sub-martingales:IE(Xt) ≥ IE(X0) for all t ≥ 0•For super-martingales:IE(Xt) ≤ IE(X0) for all t ≥ 0•If {(Xt, Ft)} is a martingale, {Gt} is a filtration, Gt⊆ Ft, and{Xt} is {Gt}-adapated, then {(Xt, Gt)} is a martingale◦Similar statements hold for sub-martingales andsuper-martingalesMATH136/STAT219 Lecture 16, October 31, 2008 – p. 6/10Zero-Mean, Independent Increment Processes•If {Xt} is adapted to filtration {Ft} with IE|Xt| < ∞ and forall 0 ≤ s ≤ t◦IE(Xt− Xs) = 0◦Xt− Xsis independent Fs•Then {(Xt, Ft)} is a martingale•Corollary: if {Wt} is a Brownian motion with canonicalfiltration {FWt} then {(Wt, FWt)} is a martingaleMATH136/STAT219 Lecture 16, October 31, 2008 – p. 7/10Functions of Sub-Martingales•Suppose {(Xt, Ft)} is a martingale and g is a convexfunction such that IE|g(Xt)| < ∞ for all t ≥ 0. Then{(g(Xt), Ft)} is a sub-martingale•Suppose {(Zt, Gt)} is a sub-martingale and h is anondecreasing, convex function such that IE|h(Zt)| < ∞ forall t ≥ 0. Then {(h(Zt), Gt)} is a sub-martingaleMATH136/STAT219 Lecture 16, October 31, 2008 – p. 8/10Right-Continuous Filtrations•Suppose {Ft} is a filtration and defineFt+.=\h>0Ft+h•Ft+is “information available immediately after time t”•A filtration is right-continuous if Ft+= Ftfor all t ≥ 0•“Each new piece of information has a definite first time ofarrival"MATH136/STAT219 Lecture 16, October 31, 2008 – p. 9/10Right-Continuous Filtrations (cont.)•Not every filtration is right-continuous (see Example 4.2.12)•However, a filtration can be augmented so it isright-continuous•For any filtration {Ft} we will always assume the “usualconditions”◦{Ft} is right-continuous◦N ∈ F0whenever “IP (N) = 0”•Consequence: if {(Xt, Ft)} is a martingale and {Ft} isright-continuous then {Xt} has an RCLL modificationMATH136/STAT219 Lecture 16, October 31, 2008 – p.
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