 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2010

Paper 29

No <quadratic variation> argument is needed. For deterministic $u<v$, the <martingale> property and square integrability give
$$
\mathbb E(X_vX_u)=\mathbb E\bigl[X_u\mathbb E(X_v\mid\mathcal F_u)\bigr]=\mathbb E X_u^2.
$$
Consequently
$$
\mathbb E(X_v-X_u)^2=\mathbb E X_v^2-\mathbb E X_u^2.
$$
Apply this identity separately to each consecutive grid increment. The second moments telescope, giving
$$
\boxed{\mathbb E V_t^{n,|\cdot|^2}=\mathbb E X_{t_n}^2-\mathbb E X_0^2
\le\sup_{s\ge0}\mathbb E X_s^2-\mathbb E X_0^2<\infty.}
$$
This uniform bound follows directly from <martingale-difference orthogonality>, including the final interval specified by the ceiling.

Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)