Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 3 / b / Solution

The defining relation gives $A_1^{(n)}=X_1^2-2M_1^{(n)}$. Applying $(r-s{)}^2\le 2r^2+2s^2$ and the bound from part (a),Thusuniformly in the dyadic mesh. This controls the approximations to quadratic variation without assuming that their limits already exist.