= Locally bounded process
A <locally bounded process> can be made bounded by stopping it on an increasing sequence of random time <intervals>. For <stochastic integral>[stochastic integration] on positive times, one useful convention is that there exist <stopping time>[stopping times] $\tau_n\uparrow\infty$ <almost surely> and finite deterministic constants $K_n$ such that
$$
|H_t|\mathbf1_{\{0<t\leq\tau_n\}}\leq K_n
\quad\text{almost surely for all }t.
$$
The <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 <adapted process>[continuous adapted process] with finite initial value has this positive-time localization. Set $\tau_n=\inf\{t\geq0:|H_t|\geq n\}\wedge n$. 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 process>[predictable] <locally bounded process> is <integrable> against a <semimartingale>: the bounded stopped integrands define compatible <integrals>, which are joined by localization. <predictable process>[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 process>[adapted] paths give the stated localization.
// Target: probability-and-statistics.bigb
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