measurability of stopped processes
Let $X$ be a realvalued stochastic process^{} and $\tau $ be a stopping time. If $X$ satisfies any of the following properties then so does the stopped process ${X}^{\tau}$.

1.
$X$ is jointly measurable.

2.
$X$ is progressively measurable.

3.
$X$ is optional.

4.
$X$ is predictable.
In particular, if $X$ is a rightcontinuous and adapted process then it is progressive (alternatively, it is optional). Then, the stopped process ${X}^{\tau}$ will also be progressive and is therefore rightcontinuous and adapted.
Also, for any progressive process $X$ and bounded stopping time $\tau \le t$, the above result shows that ${X}_{\tau}={X}_{t}^{\tau}$ will be ${\mathcal{F}}_{t}$measurable.
Title  measurability of stopped processes 

Canonical name  MeasurabilityOfStoppedProcesses 
Date of creation  20130322 18:39:00 
Last modified on  20130322 18:39:00 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  4 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 60G05 
Related topic  MeasurabilityOfStochasticProcesses 