Predictable stopping-time grid

Assume $T>0$ and work on a filtered probability space with stopping times and stopped sigma-algebras as defined in Continuous-time stopping times and stopped sigma-algebras. A finite predictable stopping-time grid on $[0,T]$ consists of an integer-valued random variable $N$ with $1\le N<\mathrm{\infty }$ almost surely and a sequence $({\tau }_n{)}_{n\ge 0}$ of stopping times. It satisfies ${\tau }_0=0$, ${\tau }_n<{\tau }_{n+1}$ on $\{n<N\}$, and ${\tau }_n=T$ on $\{n\ge N\}$. For every fixed integer $n\ge 0$, define $h_n={\tau }_{n+1}-{\tau }_n$ and require that $h_n$ is ${\mathcal{F}}_{{\tau }_n}$-measurable; on $\{n<N\}$ require also $0<h_n\le h_{\max }$ for a deterministic $h_{\max }>0$. Thus $h_n=0$ on $\{n\ge N\}$. A budgeted predictable grid has a deterministic integer $B$ with $N\le B$ almost surely.

The condition $h_n\in {\mathcal{F}}_{{\tau }_n}$ says the next interval is selected using information available at its left endpoint. Predictability is relative to the specified filtration: a step choice using a trial Brownian increment is excluded when it fails this measurability test. No Brownian motion or filtration-relative independence assumption is part of this definition. Brownian-bridge refinement can retain sampled increments, but retaining them does not by itself establish the required measurability of the step choice.