proof of measurability of stopped processes
Let be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on the measurable space , be a stopping time, and be a stochastic process. We prove the following measurability properties of the stopped process .
If is jointly measurable then so is .
Suppose first that is a process of the form for some and . Then, is -measurable. By the functional monotone class theorem, it follows that is a bounded -measurable process whenever is. By taking limits of bounded processes, this generalizes to all jointly measurable processes.
If is progressively measurable then so is .
For any given , let be the -measurable process . As is -measurable, the result proven above says that will also be -measurable, so is progressive.
If is optional then so is .
As the optional processes are generated by the right-continuous and adapted processes then it is enough to prove this result when is right-continuous and adapted. Clearly, will be right-continuous. Also, will be progressive (see measurability of stochastic processes) and, by the result proven above, it follows that is progressive and, in particular, is adapted.
If is predictable then so is .
By the definition of predictable processes, it is enough to prove the result in the cases where for some and , and where is the minimal element of and .
In the first case, is predictable and, in the second case, is predictable.
|Title||proof of measurability of stopped processes|
|Date of creation||2013-03-22 18:39:03|
|Last modified on||2013-03-22 18:39:03|
|Last modified by||gel (22282)|