properties of $X$integrable processes
Let $X$ be a semimartingale. Then a predictable process $\xi $ is $X$integrable if the stochastic integral $\int \xi \mathit{d}X$ is defined, which is equivalent^{} to the set
$$\{{\int}_{0}^{t}\alpha \mathit{d}X:\alpha \le \xi \text{is predictable}\}$$ 
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 $\lambda ,\mu \in \mathbb{R}$, 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.
Title  properties of $X$integrable processes 

Canonical name  PropertiesOfXintegrableProcesses 
Date of creation  20130322 18:40:59 
Last modified on  20130322 18:40:59 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  5 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 60H10 
Classification  msc 60G07 
Classification  msc 60H05 
Related topic  StochasticIntegration 