Let $T=\inf\{t\geq0:X_t\notin(-a,b)\}$, and suppose the jumps of $X$ have absolute value at most $c$. A nonconstant centered finite-variance <Lévy process> oscillates, so $T<\infty$ almost surely. Before $T$ the process lies in $(-a,b)$, and at $T$ its bounded overshoot gives $X_T\in[-a-c,b+c]$. Thus the variables $X_{T\wedge n}^2$ are uniformly bounded.

Apply the <optional sampling theorem for a supermartingale> to the martingale from part (c):
$$
\mathbb E[X_{T\wedge n}^2]
=\sigma^2\mathbb E[T\wedge n].
$$
<Bounded convergence theorem> on the left and <monotone convergence theorem> on the right yield
$$
\mathbb E[X_T^2]=\sigma^2\mathbb E[T].
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / d / Solution

{wiki}

On a <finite measure> space, if measurable functions are uniformly bounded and converge pointwise, then their integrals converge. This is the <dominated convergence theorem> with a constant dominating function.

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 Incoming links: Bounded convergence theorem