Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 2

The predictable sigma-algebra $\mathcal{P}$ on $\Omega \times [0,\infty )$ is generated by the sets $A\times \{0\}$ with $A\in {\mathcal{F}}_0$ and $A\times (s,t]$ with $0\le s<t$ and $A\in {\mathcal{F}}_s$. A process is previsible precisely when it is $\mathcal{P}$-measurable.

A simple predictable process has the formwhere $0=t_0<\cdots <t_m<\infty$, $H_0$ is bounded and ${\mathcal{F}}_0$-measurable, and each $H_k$ is bounded and ${\mathcal{F}}_{t_k}$-measurable. Finite linear combinations of indicators of predictable rectangles are of this form after refining the finitely many time partitions.

Those rectangles form a semiring of sets generating $\mathcal{P}$. The collection of sets whose indicators can be approximated in $L^2(\nu )$ by simple predictable processes is a monotone class: for an increasing sequence, truncate the union and use the finiteness of $\nu$; complements and finite disjoint unions are handled by linearity. The Monotone class theorem therefore puts every $\mathcal{P}$-measurable indicator in the closure. Ordinary measurable simple functions are dense in $L^2(\mathcal{P},\nu )$, so simple predictable processes are dense there as well.

Every simple predictable process is left-continuous and adapted: on $(t_k,t_{k+1}]$ its value is already ${\mathcal{F}}_{t_k}$-measurable. Hence any sigma-algebra making all left-continuous adapted processes measurable contains the generators of $\mathcal{P}$.

Conversely, let $X$ be left-continuous and adapted. For $n\ge 1$, defineOn each bounded time interval this is a simple predictable process, and left continuity gives $X_t^{(n)}\to X_t$ for every $t$. Thus $X$ is $\mathcal{P}$-measurable. This proves that $\mathcal{P}$ is exactly the smallest sigma-algebra making every left-continuous adapted process measurable.

If $X:(\Omega \times [0,\infty ),\mathcal{P})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable and $f$ is Borel measurable, then $f\circ X$ is measurable by composition of measurable functions. Hence applying a Borel function pointwise to a predictable process again gives a predictable process.

A deterministic càdlàg function is Borel measurable. The sets $\Omega \times (s,t]$ and $\Omega \times \{0\}$ show that $\mathcal{B}([0,\infty ))$ is contained in $\mathcal{P}$. Therefore every deterministic càdlàg function, regarded as a process constant in $\omega$, is predictable.

Let $Y$ be a nonconstant Bernoulli random variable, let ${\mathcal{F}}_t$ be trivial for $t<1$, and let ${\mathcal{F}}_t=\sigma (Y)$ for $t\ge 1$. This is a right-continuous filtration after completion. The processis adapted and càdlàg. If it were predictable, its value $X_1=Y$ at the deterministic time $1$ would be measurable with respect to the left-limit sigma-algebra ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{t<1}{\mathcal{F}}_t)$, which is trivial. That contradicts the choice of $Y$, so $X$ is not predictable.