Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / b

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- Solution b

Solution

Let

τ_{n}

localize the nonnegative local martingale

M

. For

s \leq t

,

E (M_{t \land τ_{n}} ∣ F_{s}) = M_{s \land τ_{n}} .

Conditional Fatou lemma and nonnegativity give

E (M_{t} ∣ F_{s}) \leq lim inf_{n} M_{s \land τ_{n}} = M_{s} .

Thus

M

is a supermartingale, recovering the general fact about a nonnegative local martingale.

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