1 What are Stopping Times?
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 \(n\).
Let \(\tau\) be a random variable which is defined on the 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 \(t\). For example:
- 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.
For discrete stochastic processes \((X_{n})\) then we have equivalent definitions for \(\{ \tau \leq n \}=\{ \tau \geq n \}^c\) and \(\{ \tau >n \}=\{ \tau \leq n \}^c\). This can easily be shown, for example for \(\{ \tau \leq t \}\) since \[ \{ \tau \leq n \}=\bigcup_{s=1}^n\{ \tau=s \}\in \mathcal{F}_{n}, \] since \(\mathcal{F}_{n}\) is closed under countable unions. However we note that in continuous time this generally doesn’t hold since \[ \{ \tau \leq t \}=\bigcup_{s \in [0,t]}\{ \tau=s \}\not\in \mathcal{F}_{t} \] generally since the union might be uncountable. However, under the additional assumption that the filtration \(\mathbb{F}\) is right-continuous, that is \[ \mathcal{F}_{t}=\bigcap_{s>t}\mathcal{F}_{s}, \] one can show that the standard definition is equivalent to:
- \(\{ \tau<t \}\in F_{t-}:=\sigma\left( \bigcup_{s<t}\mathcal{F}_{s} \right)\)
- and sometimes to conditions involving \(\{ \tau=t \}\) but with extra care.
2 ⏱ Stopping Time σ-Algebra
Next we define the \(\sigma\)-algebra for a stopping time, i.e. the information about whether a stopping time has occurred or not.
The \(\sigma\)-algebra for a stopping time \(\tau\) is defined by \[ \mathcal{F}_{\tau}=\left\{A \in \mathcal{F}: A \cap \{\tau \leq t\} \in \mathcal{F}_{t} \text{ for all } t\right\}. \] This represents the information available up to the (random) stopping time \(\tau\).
3 Stopped Processes
Once we have a stopping time \(\tau\), we can construct a stopped process by “killing” the original process at time \(\tau\).
Given a stochastic process \((X_t)_{t \in T}\) and a stopping time \(\tau\), the stopped process is defined as: \[ X_t^{\tau} = X_{t \wedge \tau} \] where \(t \wedge \tau = \min(t, \tau)\).
In other words, the process evolves normally until time \(\tau\), then remains constant afterward.
Key property: If \((X_n)\) is a martingale, then the stopped process \((X_n^{\tau})\) is also a martingale. This is essential for applying the Optional Stopping Theorem.
Example: For a [[simple-random-walk]], \(X_n^{\tau} = X_{n \wedge \tau}\) tracks the walk until it first hits a boundary, then freezes at that value.
Stopped processes are central to:
- Proving the Optional Stopping Theorem
- Martingale inequalities and concentration bounds
- Optimal stopping and sequential decision-making
4 Examples of Stopping Times
Suppose that \(T\) and \(S\) are stopping times with respect to filtration \(\mathbb{F}:=(\mathcal{F}_{n})\). We can show that the following are stopping times: (1) \(T+S\); (2) \(T\wedge S\); and (3) \(T \vee S\).
Firstly, note that \[ \{ T+S\leq t\} = \{ T \leq t-S \}\in \mathcal{F}_{t-S}\subseteq \mathcal{F}_{t}, \] thus \(T+s\) is a stopping time. Secondly, we clearly have \[ \{ T\wedge S\leq t \}=\{ T\leq t \} \cup \{ S\leq t \}\in\mathcal{F}_{t}, \] since \(\sigma\)-algebra are closed under countable unions, thus \(T \wedge S\) is a stopping time. Finally \[ \{ T\vee S\leq t \}=\{ T \leq t \}\cap \{ S \leq t \}\in \mathcal{F}_{t}, \] since \(\sigma\)-algebra are closed under countable intersections, thus \(T \vee S\) is a stopping time.
5 Sequences of Stopping Times
Assume for the following section that the filtration \(\mathbb{F}\) is complete and right-continuous. Consider the sequence of stopping times \((\tau_{n})\). Then we can show that \(\sup_{n}\tau_{n}\), \(\inf_{n}\tau_{n}\), \(\liminf_{n}{\tau_{n}}\) and \(\limsup_{n}{\tau_{n}}\) are also stopping times.
The following link on stack exchange contains proof of these results.
6 Bounded vs Unbounded Stopping Times
An important distinction in stopping time theory is whether the stopping time is bounded or unbounded.
Bounded stopping time: \(\tau \leq N\) a.s. for some finite constant \(N\). All bounded stopping times satisfy the Optional Stopping Theorem without additional conditions.
Unbounded stopping time: No finite upper bound. Unbounded stopping times require additional regularity conditions (e.g., \(\mathbb{E}[\tau] < \infty\), uniform integrability) for the Optional Stopping Theorem to apply.
This distinction matters because many theorems in probability have weaker assumptions for bounded stopping times. For instance, the Optional Stopping Theorem is automatic for bounded \(\tau\), but requires careful justification for unbounded \(\tau\).
7 Applications of Stopping Times
Stopping times appear across probability theory and applications:
- Optional Stopping Theorem - Fundamental theorem connecting stopping times to martingales
- First Passage Times / Hitting Times - When a process reaches a target set
- [[optimal-stopping-theory]] - Finding the best time to act (e.g., when to sell an asset)
- Sequential Analysis - Hypothesis testing with a random stopping rule
- Change-point Detection - Detecting when a process changes behavior
- Gambler’s Ruin & Exit Times - Analysis of random walks with boundaries
- Martingale Inequalities - Proving maximal inequalities and concentration bounds