Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 6 / c

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- Solution c

Solution

A martingale must be integrable. On the event

{N_{t} = 1}

,

X_{t} = g (Y_{1})

, so integrability of

X_{t}

\int_{0}^{1} ∣ g (y) ∣ d y < \infty.

For

s < t

, independent increments give

E [X_{t} - X_{s} ∣ F_{s}] = λ (t - s) \int_{0}^{1} g (y) d y .

Therefore the necessary and sufficient condition is

g \in L^{1} [0, 1] and \int_{0}^{1} g (y) d y = 0.

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