measurability of stopped processes

Let X be a real-valued stochastic processMathworldPlanetmath and τ be a stopping time. If X satisfies any of the following properties then so does the stopped process Xτ.

  1. 1.

    X is jointly measurable.

  2. 2.
  3. 3.

    X is optional.

  4. 4.

    X is predictable.

In particular, if X is a right-continuous and adapted process then it is progressive (alternatively, it is optional). Then, the stopped process Xτ will also be progressive and is therefore right-continuous and adapted.

Also, for any progressive process X and bounded stopping time τt, the above result shows that Xτ=Xtτ will be t-measurable.

Title measurability of stopped processes
Canonical name MeasurabilityOfStoppedProcesses
Date of creation 2013-03-22 18:39:00
Last modified on 2013-03-22 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