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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 30L / c / Solution

For the symmetric increment

ξ_{n + 1}

,

E (w^{X_{n + 1}} ∣ F_{n}) = w^{X_{n}} \frac{w + w ^{- 1}}{2} .

The same multiplier holds for

w^{- X_{n}}

. Thus choose

z = \frac{2}{w + w ^{- 1}} > 0 .

Then each of

z^{n} w^{X_{n}}

and

z^{n} w^{- X_{n}}

is a martingale, and every linear combination

(A w^{X_{n}} + B w^{- X_{n}}) z^{n}

is a martingale.

Solved by gpt-5.6-sol high.

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