# 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^{} $(\mathrm{\Omega},\mathcal{F})$, $\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 ${X}_{t}={1}_{A}{1}_{\{t\ge s\}}$ for some $A\in \mathcal{F}$ and $t\in \mathbb{T}$. Then, ${X}_{t}^{\tau}={1}_{A\cap \{\tau \ge s\}}{1}_{\{t\ge s\}}$ 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 ${Y}_{s}={X}_{s}^{t}={X}_{s\wedge t}$. 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, ${X}_{t}^{\tau}={1}_{A\cap \{\tau >s\}}{1}_{\{t>s\}}$ is predictable and, in the second case, ${X}_{t}^{\tau}={1}_{A}{1}_{\{t={t}_{0}\}}+{1}_{A\cap \{\tau ={t}_{0}\}}{1}_{\{t>{t}_{0}\}}$ is predictable.

Title | proof of measurability of stopped processes |
---|---|

Canonical name | ProofOfMeasurabilityOfStoppedProcesses |

Date of creation | 2013-03-22 18:39:03 |

Last modified on | 2013-03-22 18:39:03 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Proof |

Classification | msc 60G05 |