 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-2/30k/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The process

M

is adapted and integrable. Since

A_{n + 1}

F_{n}

-measurable,

E [M_{n + 1} ∣ F_{n}] = E [V_{n + 1} ∣ F_{n}] + A_{n + 1} = E [V_{n + 1} ∣ F_{n}] + A_{n} + V_{n} - E [V_{n + 1} ∣ F_{n}] = V_{n} + A_{n} = M_{n} .

Thus $M$ is a martingale. This is the finite-horizon Doob decomposition $V = M - A$ .

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