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 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.

• $\tau_n<\tau$ whenever $\tau>0$
• $\tau_n\rightarrow\tau$ as $n\rightarrow\infty$

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 $\{\tau\le t\}$ 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 $\tau_n\le\tau$ 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 $\mathbb{T}$