Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 202 / 2 / a / Existence of the quadratic-variation limit / Solution

By the stated Cauchy property and completeness of

M^{2}

, there is a square-integrable continuous martingale

M

such that

sup_{t \geq 0} E ∣ M_{t}^{(n)} - M_{t} ∣^{2} ⟶ 0.

A_{t} = X_{t}^{2} - M_{t}

. This process is continuous and adapted, and

E sup_{t \geq 0} ∣ A_{t}^{(n)} - A_{t} ∣^{2} = E sup_{t \geq 0} ∣ M_{t}^{(n)} - M_{t} ∣^{2} \leq 4 sup_{t \geq 0} E ∣ M_{t}^{(n)} - M_{t} ∣^{2} ⟶ 0

by the Doob L2 maximal inequality. The process

A

is the quadratic variation

[X]

.

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