Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 3 / b

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- Solution b

Solution

By the tower property of conditional expectation and the defining martingale identity,

E [M_{n}] = E [E [M_{n} ∣ F_{n - 1}]] = E [M_{n - 1}] .

Iteration gives

E [M_{n}] = E [M_{0}] (0 \leq n \leq N) .

The integrability in the definition ensures these expectations exist.

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