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By van der Vaart A.W.

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Some authors always use the natural filtration, or its completion. Property (iii) is stronger if {Ft } is a bigger filtration. Brownian motion is “the” example of a continuous martingale. 26 Theorem. Any Brownian motion is a martingale. Proof. Because Bt − Bs is independent of Fs , we have E(Bt − Bs | Fs ) = E(Bt − Bs ) almost surely, and this is 0. 27 EXERCISE. Show that the process {Bt2 − t} is a martingale. Brownian motion has been studied extensively and possesses many remarkable properties.

Iii) The process (Nt − t)2 − t is a martingale. 9 EXERCISE. Show that every cadlag super martingale is right contin- uous in mean. 3 37 Martingale Convergence The martingale convergence theorems for discrete time martingales extend without surprises to the continuous time situation. 10 Theorem. If X is a uniformly integrable, cadlag (sub/super) mar- tingale, then there exists an integrable random variable X∞ such that Xt → X∞ almost surely and in L1 as t → ∞. s. for all t ≥ 0. s. for t ≥ 0. Furthermore, if X is Lp -bounded for some p > 1, then Xt → X∞ also in Lp .

This is the case, for instance, if the sample paths are continuous, or more generally left- or right continuous. Most of the stochastic processes that we shall be concerned with possess this property. 1: Stochastic Processes 33 often consider cadlag processes (from “continu ` a droite, limite ` a gauche”): processes with sample paths that are right-continuous and have limits from the left at every point t > 0. If X is a left- or right-continuous process, then Xt− = lim Xs , s↑t,st define left- and right-continuous processes.

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