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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

For an elementary previsible process $H={\sum }_{j=0}^{r-1}{H}_j{1}_{(t_j,t_{j+1}]}$, with bounded ${\mathcal{F}}_{t_j}$-measurable coefficients, defineThe Itô isometry isMore generally the squared $L^2$ distance between the integrals of $H$ and $K$ is the integral of $(H-K{)}^2$ on the right. The integral is a zero-starting continuous square-integrable martingale.

Elementary predictable processes are dense in $L^2(M)$. Given $H$ in this space, choose elementary $H_n$ converging in its norm and define the Itô integral by the $L^2$ limit of $H_n\cdot M$. The isometry proves that this is independent of the approximating sequence and remains valid for the limit. The Doob L2 maximal inequality bounds the expected squared path supremum of the difference by four times its terminal second moment; hence the convergence also gives a continuous martingale version, with uniform convergence on compacts in probability.

For a continuous local martingale $M$ and locally bounded process $H$ that is previsible, choose increasing stopping times ${\tau }_n\uparrow \infty$ such that $M^{{\tau }_n}$ is L2-bounded, $|H|\le K_n$ up to ${\tau }_n$, and ${\tau }_n\le n$. This can be done by combining localizers with first exits of $M$ from bounded intervals and the local bounds of $H$. Then $H{1}_{[0,{\tau }_n]}$ is in the integrand space for $M^{{\tau }_n}$. Define the integral on each stopped interval by the preceding construction. The stopping property of elementary integrals, extended by the isometry, makes these definitions agree on overlaps. Pasting givesThis is a localization construction; global square integrability of the resulting process is not asserted.