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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 30K / c / Solution

The process

M

is adapted and integrable. Since

A_{n + 1}

is

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

.

Solved by gpt-5.6-sol high.

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