predictable stopping time
A predictable, or previsible stopping time is a random time which is possible to predict just before the event. Letting ${({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}}$ be a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space^{} $(\mathrm{\Omega},\mathcal{F})$, then, a stopping time $\tau $ is predictable if there exists an increasing sequence of stopping times ${\tau}_{n}$ satisfying the following.

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$$ whenever $\tau >0$.

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${\tau}_{n}\to \tau $ as $n\to \mathrm{\infty}$.
The sequence ${\tau}_{n}$ is said to announce or foretell $\tau $.
For example, if $X$ is a continuous^{} adapted process with ${X}_{0}=0$, such as Brownian motion^{}, then the first time $\tau $ at which it hits a given level $K\ne 0$ is a predictable stopping time. In this case, if ${\tau}_{n}$ is the first time at which $X$ hits the level $K(11/n)$, then the sequence ${\tau}_{n}$ announces $\tau $.
On the other hand, if $X$ is a Poisson process then the first time $\tau $ at which it is nonzero is not predictable. To show this, suppose that $$ are stopping times. The fact that ${X}_{t}\lambda t$ is a martingale^{} means that Doob’s optional sampling theorem^{} can be applied, giving $\mathbb{E}[{X}_{{\tau}_{n}}\lambda {\tau}_{n}]=0$. Then, ${X}_{t}=0$ for $$ gives $\mathbb{E}[{\tau}_{n}]=0$. So, ${\tau}_{n}=0$ with probability one, and the sequence ${\tau}_{n}$ cannot announce $\tau $.
In discrete time, where the filtration $({\mathcal{F}}_{t})$ has time $t$ running over the index set^{} ${\mathbb{Z}}_{+}$, then a stopping time is said to be predictable if $\{\tau \le t\}$ is ${\mathcal{F}}_{t1}$measurable for every time $t=1,2,\mathrm{\dots}$.
This can be generalized to an arbitrary index set $\mathbb{T}$, where a stopping time $\tau :\mathrm{\Omega}\to \mathbb{T}\cup \{\mathrm{\infty}\}$ is predictable if there exists an increasing sequence of stopping times ${\tau}_{n}\le \tau $ such that $$ whenever $\tau $ is not equal to a minimal element of $\mathbb{T}$, and ${\bigcap}_{n}({\tau}_{n},\tau )$ contains no elements of $\mathbb{T}$.
Title  predictable stopping time 
Canonical name  PredictableStoppingTime 
Date of creation  20130322 18:37:19 
Last modified on  20130322 18:37:19 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  6 
Author  gel (22282) 
Entry type  Definition 
Classification  msc 60G40 
Classification  msc 60G05 
Synonym  predictable time 
Synonym  previsible time 
Synonym  previsible stopping time 
Related topic  StoppingTime 
Related topic  PredictableProcess 