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By Kai Lai Chung

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Extra info for Lectures from Markov Processes to Brownian Motion

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0 Two stochastic processes X = {X t} and Y = {1';} are said to be versions of each other iff we have 'it: P{Xt = 1';} = L (14) lt then follows that for any countable subset S of T, we have also P{X t = 1'; forall tES} = L In particular, for any (t h .. ,,(,,), the distributions of (X tl' . . ,XtJ and (1';" ... , Y,J are the same. Thus the two processes X and Y have identical finite-dimensional joint distributions. 4. Martingale Theorems set. We shall not delve into this question but proceed to find a good version for a supermartingale, under certain conditions.

Recall that for a discrete time positive supermartingale, almost surely every sampie sequence remains at the value zero if it is ever taken. In continuous time the result has a somewhat delicate ramification. Theorem 4. Let {X t, ffr} be a positive supermartingale having right continuous paths. Let T 1(w) = inf{t;;::-: 0IXiw) = O}, T 2 (w) = inf{t;;::-: 0IXt-(w) = O}, T= T I /\ T 2 • Then we have almost surely X(T + t) = ° Jor all t;;::-: ° on the set {T < oo}. ProoJ. 4. By Theorem 1, both {Xt} and {X t -} are progressively 42 1.

4. 0B x %0 then for each A E tff', the function (t,x)--+ Pt(x, A) is in gg x tff'. Hence this is the case if {X t} is right [or left] con- tinuous. [Hint: consider the dass of functions ((J on T x Q such that (t,x)--+P{((J} belongs to fJ6xtff'. ] 5. If {X t } is adapted to {~}, and progressively measurable relative to {~+ ,} for each 8 > 0, then {X t } is progressively measurable relative to f%} l t· 6. g;;;} but not progressively measurable relative to {~}. 7. Suppose that {~} is right continuous and Sand T are optional relative to {~}, with S s T.

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