Compare the (last part of the) first ("pedestrian") proof of Theorem 6.5 with the characterization of a BM by Lemma 5.4. Can you think of a way to use this lemma instead of the direct calculation?

Data From Theorem 6.5

Data From Lemma 5.4

Transcribed Image Text:

6.5 Theorem (strong Markov property). Let (Bt)to be a BMd with admissible filtration (F)to and a finite stopping time, i.e. (w) < co for all we Q. Then (W)tao, where W = Bo+t - Bo, is again a BMd, which is independent of Fo+. We will give two proofs of this important result, first a down-to-earth direct attack, and then a second one which relies on an ingenious martingale-plus-optional-sampling trick.