 Locally bounded process

A locally bounded process can be made bounded by stopping it on an increasing sequence of random time intervals. For stochastic integration on positive times, one useful convention is that there exist stopping times ${\tau }_n\uparrow \infty$ almost surely and finite deterministic constants $K_n$ such thatThe interval convention matters at time zero: including zero additionally requires a deterministic bound on $H_0$, whereas the positive-time Stochastic integral does not use that value.

Every continuous adapted process with finite initial value has this positive-time localization. Set ${\tau }_n=\inf \{t\ge 0:|H_t|\ge n\}\wedge n$. If its initial magnitude already exceeds $n$, this Stopping time is zero and the indicated interval is empty. Otherwise continuity prevents overshoot at the first exit, so $K_n=n$ works. continuity on each compact time interval makes ${\tau }_n\to \infty$ pathwise.

A predictable locally bounded process is integrable against a semimartingale: the bounded stopped integrands define compatible integrals, which are joined by localization. Predictability supplies the required information condition; a magnitude bound alone does not replace it. For example, Brownian motion is not bounded on the whole infinite time axis but its continuous adapted paths give the stated localization.

// Target: probability-and-statistics.bigb