Brownian Motion and Martingales in Analysis (Wadsworth & by Richard Durrett

By Richard Durrett

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2, treating the jumps properly is a delicate matter, (ii) in all our applications the local martingales are continuous, and (iii) the assumption of continuity allows us to considerably simplify many of the main proofs and formulas of the theory. ). F,). 14) that the Brownian filtration has this property. The last result is interesting in its own right and would be in the book in any case, so I don't think this is too great a price to pay for the enormous simplifications that result. The, following is an example of the simplifications mentioned above: (2) When X has continuous paths, we can always take T.

T. ,ds=supE f Hsds J = r ,Jo f The definitions above should suggest our strategy for extending the integral from Al to A2: We will pick a sequence H"ebA1 so that II H" - HIIB 0, and we will show that H" B converges to a limit which is independent of the sequence of approximations chosen. To carry out the first step in this program, we need to show (7) If HE A2, then there is a sequence H" e bA1 so that II H" - HII B -. 0. m GtIIB _+ 0, so G-'([2-t]12-). As t -+ oo, IIG" if we let H" = G",'- and m" -+ oc fast enough, II H" - HIIB -.

One of the reasons we have written out the last proof in such great detail is to show its dependence on the fact that K is closed. (2) implies that if T = inf{ t > 0 : B, E An} is a stopping time and An T A, then T = inf {t > 0 : B, E A} is also ; but to go the other way, that is, let An A and conclude Tn T T, we need to know that A is closed. The last annoying fact makes it difficult to show that the hitting time of a Borel set is a stopping time, and, in fact, this is not true unless the a-fields are completed in a suitable way.

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