 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-25/3/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

The defining relation gives

A_{1}^{(n)} = X_{1}^{2} - 2 M_{1}^{(n)}

. Applying

(r - s)^{2} \leq 2 r^{2} + 2 s^{2}

and the bound from part (a),

E (A_{1}^{(n)})^{2} \leq 2 E X_{1}^{4} + 8 E (M_{1}^{(n)})^{2} \leq 2 C^{4} + 8 C^{4} .

Thus

E (A_{1}^{(n)})^{2} \leq 10 C^{4},

uniformly in the dyadic mesh. This controls the approximations to quadratic variation without assuming that their limits already exist.

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