Show that Wt and are local martingales, where W is defined in the statement of Theorem 12.2. Suppose {Ft} is a filtration satisfying the usual conditions, X is a Brownian motion with respect to...

Show that W_t
and

are local martingales, where W is defined in the statement of Theorem 12.2.

Suppose {F_t} is a filtration satisfying the usual conditions, X is a Brownian motion with respect to {F_t}, and T is a finite stopping time with respect to this same filtration. Let Y be another Brownian motion that is independent of {F_t} and define

Show that Z is a Brownian motion (although not necessarily with respect to {F_t}).