# 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

 $\left\{\int_{0}^{t}\alpha\,dX:|\alpha|\leq|\xi|\textrm{ is predictable}\right\}$

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

1. 1.

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

2. 2.

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

3. 3.

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

4. 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 $1_{\{t\leq\tau_{n}\}}\xi_{t}$ is $X$-integrable, then $\xi$ is $X$-integrable.

Title properties of $X$-integrable processes PropertiesOfXintegrableProcesses 2013-03-22 18:40:59 2013-03-22 18:40:59 gel (22282) gel (22282) 5 gel (22282) Theorem msc 60H10 msc 60G07 msc 60H05 StochasticIntegration