Let

τ_{n}

localize

M

E M_{t \land τ_{n}}^{2} = E [M]_{t \land τ_{n}} \leq E [M]_{t} .

Thus the stopped variables are bounded in

L^{2}

, and localization plus weak compactness shows that

M

is a true martingale. The Itô formula applied to

f (x) = x^{2}

gives

M_{t}^{2} - [M]_{t} = 2 \int_{0}^{t} M_{s} d M_{s} .

E [M]_{t} < \infty

supply the required local integrability, so it is a true martingale.

2

Solution

Part 1 gives

E M_{t}^{2} = E [M]_{t}

. Hence

sup_{t \geq 0} E M_{t}^{2} \leq E [M]_{\infty} < \infty,

M

L^{2}

-bounded.

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