proof of measurability of stopped processes

Let (t)t𝕋 be a filtrationPlanetmathPlanetmath ( on the measurable spaceMathworldPlanetmathPlanetmath (Ω,), τ be a stopping time, and X be a stochastic processMathworldPlanetmath. We prove the following measurability properties of the stopped process Xτ.

If X is jointly measurable then so is Xτ.

Suppose first that X is a process of the form Xt=1A1{ts} for some A and t𝕋. Then, Xtτ=1A{τs}1{ts} is (𝕋)-measurable. By the functional monotone class theorem, it follows that Xτ is a boundedPlanetmathPlanetmathPlanetmath (𝕋)-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 Xτ.

For any given t𝕋, let Y be the (𝕋)t-measurable process Ys=Xst=Xst. As τt is t-measurable, the result proven above says that (Xτ)t=Yτt will also be (𝕋)t-measurable, so Xτ is progressive.

If X is optional then so is Xτ.

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, Xτ will be right-continuous. Also, X will be progressive (see measurability of stochastic processes) and, by the result proven above, it follows that Xτ is progressive and, in particular, is adapted.

If X is predictable then so is Xτ.

By the definition of predictable processes, it is enough to prove the result in the cases where Xt=1A1{t>s} for some s𝕋 and As, and Xt=1A1{t=t0} where t0 is the minimal element of 𝕋 and At0.

In the first case, Xtτ=1A{τ>s}1{t>s} is predictable and, in the second case, Xtτ=1A1{t=t0}+1A{τ=t0}1{t>t0} 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