Prove that if M is a sub martingale such that the paths of M are continuous, supt is integrable, and S ≤ T are finite stopping times, then Note that the last part of the proof of Proposition 9.3...

Prove that if M is a sub martingale such that the paths of M are continuous, sup_t

is integrable, and S ≤ T are finite stopping times, then

Note that the last part of the proof of Proposition 9.3 breaks down here.

Suppose Mt is a local martingale with continuous paths. Show that if

then M^N
is a uniformly integrable martingale.