properties of $X$-integrable processes

Let $X$ be a semimartingale. Then a predictable process $\xi$ is $X$-integrable if the stochastic integral $\int \xi 𝑑X$ is defined, which is equivalent to the set

$$\{{\int }_0^t\alpha 𝑑X:|\alpha |\le |\xi |\text{ is predictable}\}$$

being bounded in probability, for each $t>0$. We list some properties of $X$-integrable processes.

1. Every locally bounded predictable process is $X$-integrable.

2. The $X$-integrable processes are closed under linear combinations. That is, if $\alpha ,\beta$ are $X$-integrable and $\lambda ,\mu \in ℝ$, then $\lambda \alpha +\mu \beta$ is $X$-integrable.

3. If $|\alpha |\le |\beta |$ are predictable processes and $\beta$ is $X$-integrable, then so is $\alpha$.

4. A process is $X$-integrable if it is locally $X$-integrable. That is, if there are stopping times ${\tau }_n$ almost surely increasing to infinity and such that $1_{\{t\le {\tau }_n\}}{\xi }_t$ is $X$-integrable, then $\xi$ is $X$-integrable.