Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 6 / a

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- Solution a

Solution

The Martingale convergence theorem states that a discrete-time martingale

(X_{n})

with uniform integrability has an integrable random variable

X_{\infty}

such that

X_{n} ⟶ X_{\infty}

almost surely and in

L^{1}

. Moreover, the martingale is closed by its limit:

X_{n} = E [X_{\infty} ∣ F_{n}] .

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