Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 1

The L2 martingale convergence theorem gives $M_t\to M_{\infty }$ in $L^2$, soConversely, apply the Doob L2 maximal inequality on $[0,T]$:Monotone convergence as $T\to \infty$ givesThus the two norms are equivalent.

For a simple predictable processdefineEach summand is a bounded predictable multiple of a martingale increment, so conditional expectation proves that $H\cdot M$ is a martingale. Orthogonality of disjoint martingale increments givesIt is therefore an $L^2$-bounded continuous martingale.

Indicators of the rectanglestogether with $A\times \{0\}$ generate the predictable sigma-algebra. Their finite linear span is precisely the set of simple processes. The monotone-class theorem therefore makes this span dense among bounded predictable functions in measure. Since $\nu$ is finite, truncation followed by bounded approximation proves density in $L^2(\mathcal{P},\nu )$.

Let ${\mathcal{M}}_c^2$ be the continuous $L^2$-bounded martingales starting at zero, modulo indistinguishability, with norm $\Vert N{\Vert }_{{\mathcal{M}}^2}=\Vert N_{\infty }{\Vert }_2$. For a fixed $M\in {\mathcal{M}}_c^2$, define a finite measure on $\mathcal{P}$ byand let $L^2(M)=L^2(\mathcal{P},{\nu }_M)$. The Itô isometry is the isometric extensionsatisfying

For the simple process in part b, orthogonality gives the sum there. Conditional on ${\mathcal{F}}_{t_i}$, the martingale identity for $M^2-[M]$ givesSumming proves the isometry. Part c then supplies the unique extension to all of $L^2(M)$.