 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-2/30k/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The process is adapted because

X_{0}

F_{0}

-measurable,

A_{k}

F_{k - 1}

-measurable, and

M_{k} - M_{k - 1}

F_{k}

-measurable. Using the predictability of

A_{n}

and the martingale increment property,

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

The assumed integrability therefore proves that

X

M

A

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