Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 1 / a

Use the predictable sigma-algebra generated by $A\times \{0\}$, $A\in {\mathcal{F}}_0$, and $A\times (s,t]$, $A\in {\mathcal{F}}_s$. A previsible process is a measurable function for this sigma-algebra. Set $J_t=1_{\{S<t\le T\}}$, with $J_0=0$. The events $\{S<t\}$ and $\{T<t\}$ are in ${\mathcal{F}}_t$, because they are countable unions of events $\{S\le q\}$ and $\{T\le q\}$ for rational $q<t$. Thus $J$ is an adapted process. Each path is left-continuous, including at the two endpoints, and is zero if $S=T$.

For a direct measurability argument, letEach summand is measurable for the predictable sigma-algebra, since its coefficient is ${\mathcal{F}}_{k{2}^{-n}}$-measurable. At every $t>0$ the sampling time approaches $t$ strictly from below, so left-continuity gives $J_t^{(n)}\to J_t$. ThereforeThe original PDF has a right-closed interval. The open right endpoint in the converted TeX is a transcription error; replacing $(S,T]$ by $(S,T)$ is not valid for arbitrary stopping times.