Let Wt be a Brownian motion and be the minimal augmented filtration generated by W . Let Show that T is not a stopping time with respect to {Ft}. Let W and S be as in Exercise 4.1. (1) Let 0 ...

Let W_t
be a Brownian motion and

be the minimal augmented filtration generated by W . Let

Show that T is not a stopping time with respect to {F_t}.

Let W and S be as in Exercise 4.1.

(1) Let 0 <><><><>

Show that there exists a constant c, depending on s,t, and u, but not a or b, such that

(2) Show that the path of a Brownian motion does not take on the same value as a local maximum twice. That is, if S and T are times when W has a local maximum, then W_S
= WT , a.s.