measurability of stochastic processes

For a continuous-time stochastic process adapted to a given filtration ( (t)t+ on a measurable spaceMathworldPlanetmathPlanetmath (Ω,), there are various conditions which can be placed either on its sample paths or on its measurability when considered as a function from +×Ω to . The following theorem lists the dependencies between these properties.


Let (Xt)tR+ be a real valued stochastic processMathworldPlanetmath. Then, X is optional if it is adapted and right-continuous, it is predictable if it is adapted and left-continuous. Furthermore, each of the following properties implies the next.

  1. 1.

    X is predictable.

  2. 2.

    X is optional.

  3. 3.

    X is progressive.

  4. 4.

    X is adapted and jointly measurable.

Title measurability of stochastic processes
Canonical name MeasurabilityOfStochasticProcesses
Date of creation 2013-03-22 18:37:29
Last modified on 2013-03-22 18:37:29
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 60G05
Related topic MeasurabilityOfStoppedProcesses