 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/30k/c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Assume (i), and let

τ

be a stopping time. The increments of the stopped process satisfy

M_{(n + 1) \land τ} - M_{n \land τ} = 1_{{τ > n}} (M_{n + 1} - M_{n}) .

Because

{τ > n} \in F_{n}

F_{n}

makes the right-hand side zero. Integrability follows from

M_{n \land τ} = M_{0} + \sum_{k = 1}^{n} 1_{{τ \geq k}} (M_{k} - M_{k - 1}),

M^{τ}

is a martingale, proving (ii).

 New to topics? Read the docs here!