Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 36 / 2 / a / Solution

The previsible sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is generated by left-continuous adapted processes. Equivalently it is generated by $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,t]$ with $A\in {\mathcal{F}}_s$, $0\le s<t$. A previsible process is a process measurable for this sigma-algebra; predictable is a synonym.

Use the zero-starting convention for ${\mathcal{M}}_c^2$: continuous martingales with $M_0=0$ and ${\sup }_t\mathbb{E}{M}_t^2<\infty$. For such an L2-bounded continuous martingale, the square-integrable stochastic integrand space isIdentify processes that agree almost everywhere for the quadratic-variation measure ${\nu }_M(d\omega ,ds)=\mathbb{P}(d\omega )d[M{]}_s$. This makes the displayed norm an actual Hilbert-space norm on equivalence classes. On a fixed horizon $[0,T]$, replace infinity by $T$; this is the same construction for continuous martingales square-integrable on that horizon. Nonzero initial values may be subtracted because the integral depends only on increments.