Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 1 / d

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- Solution d

Solution

No quadratic variation argument is needed. For deterministic

u < v

, the martingale property and square integrability give

E (X_{v} X_{u}) = E [X_{u} E (X_{v} ∣ F_{u})] = E X_{u}^{2} .

Consequently

E (X_{v} - X_{u})^{2} = E X_{v}^{2} - E X_{u}^{2} .

Apply this identity separately to each consecutive grid increment. The second moments telescope, giving

E V_{t}^{n, ∣ \cdot ∣^{2}} = E X_{t_{n}}^{2} - E X_{0}^{2} \leq s \geq 0 sup E X_{s}^{2} - E X_{0}^{2} < \infty.

This uniform bound follows directly from martingale-difference orthogonality, including the final interval specified by the ceiling.

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