proof of measurability of stopped processes

Let $(\mathcal{F}_{t})_{t\in\mathbb{T}}$ be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on the measurable space $(\Omega,\mathcal{F})$, $\tau$ be a stopping time, and $X$ be a stochastic process. We prove the following measurability properties of the stopped process $X^{\tau}$.

If $X$ is jointly measurable then so is $X^{\tau}$.

Suppose first that $X$ is a process of the form $X_{t}=1_{A}1_{\{t\geq s\}}$ for some $A\in\mathcal{F}$ and $t\in\mathbb{T}$. Then, $X^{\tau}_{t}=1_{A\cap\{\tau\geq s\}}1_{\{t\geq s\}}$ is $\mathcal{B}(\mathbb{T})\otimes\mathcal{F}$-measurable. By the functional monotone class theorem, it follows that $X^{\tau}$ is a bounded $\mathcal{B}(\mathbb{T})\otimes\mathcal{F}$-measurable process whenever $X$ is. By taking limits of bounded processes, this generalizes to all jointly measurable processes.

If $X$ is progressively measurable then so is $X^{\tau}$.

For any given $t\in\mathbb{T}$, let $Y$ be the $\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_{t}$-measurable process $Y_{s}=X^{t}_{s}=X_{s\wedge t}$. As $\tau\wedge t$ is $\mathcal{F}_{t}$-measurable, the result proven above says that $(X^{\tau})^{t}=Y^{\tau\wedge t}$ will also be $\mathcal{B}(\mathbb{T})\otimes\mathcal{F}_{t}$-measurable, so $X^{\tau}$ is progressive.

If $X$ is optional then so is $X^{\tau}$.

As the optional processes are generated by the right-continuous and adapted processes then it is enough to prove this result when $X$ is right-continuous and adapted. Clearly, $X^{\tau}$ will be right-continuous. Also, $X$ will be progressive (see measurability of stochastic processes) and, by the result proven above, it follows that $X^{\tau}$ is progressive and, in particular, is adapted.

If $X$ is predictable then so is $X^{\tau}$.

By the definition of predictable processes, it is enough to prove the result in the cases where $X_{t}=1_{A}1_{\{t>s\}}$ for some $s\in\mathbb{T}$ and $A\in\mathcal{F}_{s}$, and $X_{t}=1_{A}1_{\{t=t_{0}\}}$ where $t_{0}$ is the minimal element of $\mathbb{T}$ and $A\in\mathcal{F}_{t_{0}}$.

In the first case, $X^{\tau}_{t}=1_{A\cap\{\tau>s\}}1_{\{t>s\}}$ is predictable and, in the second case, $X^{\tau}_{t}=1_{A}1_{\{t=t_{0}\}}+1_{A\cap\{\tau=t_{0}\}}1_{\{t>t_{0}\}}$ is predictable.

Title proof of measurability of stopped processes ProofOfMeasurabilityOfStoppedProcesses 2013-03-22 18:39:03 2013-03-22 18:39:03 gel (22282) gel (22282) 4 gel (22282) Proof msc 60G05