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 $(\Omega,\mathcal{F})$ , 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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$\tau_{n}<\tau$ whenever $\tau>0$ .
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$\tau_{n}\rightarrow\tau$ as $n\rightarrow\infty$ .

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 $K\not=0$ is a predictable stopping time. In this case, if $\tau_{n}$ is the first time at which $X$ hits the level $K(1-1/n)$ , 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 $\tau_{n}<\tau$ 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 $t<\tau$ 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 $\{\tau\leq t\}$ is $\mathcal{F}_{t-1}$ -measurable for every time $t=1,2,\ldots$ .

This can be generalized to an arbitrary index set $\mathbb{T}$ , where a stopping time $\tau\colon\Omega\rightarrow\mathbb{T}\cup\{\infty\}$ is predictable if there exists an increasing sequence of stopping times $\tau_{n}\leq\tau$ such that $\tau_{n}<\tau$ 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	2013-03-22 18:37:19
Last modified on	2013-03-22 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