5. (15 pts) Let M be a continuous local martingale. For any stopping time T P( max [M+| > () S E(8 A [M]T) + P([M] T 2 8), OKt 0. Here [M] denotes the quadratic variation of M. Note that this O implies that for a sequence M" of local martingales NO [M"] -+ 0, in probability as n -+ 00 implies that max MY| - 0 in probability as n - 0. OStET Hint: Think about localizing, using a well-known inequality for sub-martingales, and the Doob-Meyer decomposition