0.1 Stopping Time
In probability theory, a stopping time is a specific type of random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest.
A stopping time is often defined by a stopping rule, which determines whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time.
The key property of a stopping rule is that is must be measurable with respect to the filtration i.e. we must know whether the rule is satisfied or not with information available at the present time \(n\).
Examples: - Turn off the freeway at the exit after the airport - This is a stopping time because the rule is based on information available at present time \(t\). - Turn off the freeway at the exit before the airport - This is not a stopping time because the rule is based on future information, we only know the exit was before the airport when we get to the airport exit.
Let \(\tau\) be a random variable which is defined on the [[knowledge-mathematics-analysis-probability-theory-probability-spaces|probability space]] \((\Omega, \mathcal{F}, \mathbb{P})\) with filtration \((\mathcal{F}_{t})_{t \in T}\) with values in \(T\) (where in most cases \(T=[0,+\infty)\)). Then \(\tau\) is a stopping time with respect to filtration \(\mathbb{F}=(\mathcal{F}_{t})_{t \in T}\) if the following condition holds \[ \{ \tau\leq t \}\in\mathcal{F}_{t}, \] for all \(t \in T\).
Intuitively, we know whether or not the stopping time \(\tau\) has occurred before or at \(t\) using only the information available at \(t\).
The \(\sigma\)-algebra for a stopping time \(T\) is defined by \[ \mathcal{F}_{T}=\left\{ A:A \cap \{ T \leq n \}~\forall n \right\}. \]