Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 24 / 1

Write the predictable sigma-algebra on positive times asIt is equivalently the smallest sigma-algebra making all left-continuous adapted processes measurable. Indeed, the indicator of each generating rectangle is left-continuous and adapted, and approximating any left-continuous adapted process by its values at the preceding points of deterministic partitions gives the reverse inclusion. Previsibility records information available immediately before observation. If time zero is included, add $B\times \{0\}$ with $B\in {\mathcal{F}}_0$. The displayed description in part (b) uses positive times, as indicated by its product space.

Observing $X_s$ reveals exactly whether $\omega <s$. Thus the natural filtration isEvery generating set either contains the entire tail $[t,\infty )$ or avoids it, so this tail remains one indistinguishable atom. Conversely, these initial intervals generate the full relative Borel sigma-algebra on $(0,t)$: use rational endpoints below $t$, together with $(0,t)$ itself. Therefore every Borel set contained in $(0,t)$, and its complement, belongs to ${\mathcal{F}}_t$. We obtainThe strict endpoint is essential: the value at $\omega =t$ is still $X_t=1$, so it cannot be distinguished from a later lifetime.

Put $D=\{(\omega ,t):\omega \ge t\}$ and let $\mathcal{C}$ denote the proposed family of sets. This is a sigma-algebra: complements replace $T$ and $A$ by their complements, and countable unions replace them by their unions. For a generating predictable rectangle $B\times (s,u]$, part (a) says either $B\subseteq (0,s)$ or $B^c\subseteq (0,s)$. In the first case its trace on $D$ is empty; in the second it is exactly $D\cap (\Omega \times (s,u])$. Its trace on $D^c$ is always product-Borel. Hence $\mathcal{P}\subseteq \mathcal{C}$.

For the reverse inclusion, $X$ is left-continuous and adapted, so $D=\{X=1\}$ belongs to the predictable sigma-algebra. Every deterministic Borel set in time also belongs to that sigma-algebra, hence $D\cap (\Omega \times T)$ is predictable. For a product rectangle $E\times T$,Each term is predictable because $E\cap (0,s)\in {\mathcal{F}}_s$. Every pair with $\omega <t$ lies in such a term by choosing $\omega <s<t$. The class of product-Borel sets $A$ for which $A\cap D^c$ is predictable is a sigma-algebra containing the product rectangles, so it contains $\mathcal{F}\otimes \mathcal{F}$. ThusThis survival-observation predictable sigma-algebra expresses a sharp distinction: before and at the lifetime the only observable coordinate is time, whereas strictly after it the lifetime is known.