Fact-checked by Grok 2 weeks ago

Stopping time

In and stochastic processes, a stopping time (also known as an optional time or Markov time) is a type of random time associated with a filtered (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, P), where the occurrence of the time is adapted to the filtration, meaning it depends only on information available up to that point without anticipating the future. In the discrete-time case, it is a non-negative integer-valued random variable \tau such that \{\tau = n\} \in \mathcal{F}_n for each n \geq 0; in continuous time, \{\tau \leq t\} \in \mathcal{F}_t for each t \geq 0. This ensures stopping times model realistic decision processes in random environments, such as first passage times \tau = \min\{n \geq 0 : X_n \in A\} (discrete) or analogous infima in continuous settings, and hitting times in Markov chains. Filtrations (\mathcal{F}_t)_{t \geq 0} form an increasing family of \sigma-algebras representing accumulating information, with \mathcal{F}_0 \subseteq \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \cdots \subseteq \mathcal{F} for s < t. Key properties include: if \tau is a , then so are \{\tau \leq t\} and the supremum or infimum of a sequence of stopping times; for stopping times S \leq T, \mathcal{F}_S \subseteq \mathcal{F}_T. These enable composition in stochastic models. Stopping times are central to martingale theory, where the optional stopping theorem preserves expectations under conditions like bounded increments or E[\tau] < \infty, aiding convergence proofs and gambling fairness. Wald's equation for i.i.d. \{X_n\} with finite mean E(X) and finite E(\tau) gives E\left(\sum_{n=1}^\tau X_n\right) = E(\tau) E(X), with uses in renewal theory and risk assessment. Applications include optimal stopping in finance (e.g., asset selling decisions), sequential testing in statistics, queueing theory for buffer exit times, and Markov chain analysis via the strong Markov property, where the process restarts independently post-\tau. These tools address uncertainty in operations research, economics, and physics.

Definitions

Discrete Time

In discrete time, a filtration is defined as an increasing sequence of σ-algebras \{\mathcal{F}_n\}_{n \geq 0} on a probability space (\Omega, \mathcal{F}, P), where \mathcal{F}_n \subseteq \mathcal{F}_{n+1} for all n, representing the progressively available information up to time n. A stopping time \tau with respect to this filtration is a random variable taking values in the extended non-negative integers \{0, 1, 2, \dots\} \cup \{\infty\} such that, for every n \geq 0, the event \{\tau \leq n\} belongs to \mathcal{F}_n. This formulation, originating in the foundational work of , captures the idea that the decision to stop by time n depends only on the information revealed up to that time, without anticipating future outcomes. An equivalent characterization is that \{\tau = n\} \in \mathcal{F}_n for all n \geq 0. This follows directly from the increasing property of the filtration, as \{\tau = n\} = \{\tau \leq n\} \setminus \{\tau \leq n-1\} (with the convention that \{\tau \leq -1\} = \emptyset), so each exact stopping event at n is \mathcal{F}_n-measurable. Stopping times may attain the value \infty with positive probability, allowing for the possibility that stopping never occurs. However, in many theoretical contexts, such as the analysis of or , it is standard to assume P(\tau < \infty) = 1, ensuring that stopping happens almost surely in finite time. The defining condition guarantees that \tau is "decided" using only information up to time n: on the event \{\tau = n\}, which is \mathcal{F}_n-measurable, the realization of \tau requires no knowledge beyond \mathcal{F}_n, as the complement \{\tau > n\} = \Omega \setminus \{\tau \leq n\} is also \mathcal{F}_n-measurable. This adaptivity is evident in the construction of stopped processes, where indicators like $1_{\{\tau \leq n\}} remain \mathcal{F}_n-measurable.

Continuous Time

In continuous-time stochastic processes, the underlying probability space is equipped with a filtration (\mathcal{F}_t)_{t \geq 0}, which is an increasing family of sub-σ-algebras of the overall σ-algebra \mathcal{F} on \Omega, representing the evolution of available information over time. This filtration is typically assumed to be right-continuous, satisfying \mathcal{F}_t = \bigcap_{s > t} \mathcal{F}_s for each t \geq 0, along with completeness (augmented by null sets). Right-continuity ensures that the filtration includes all limits of events from times approaching t from the right, which is crucial for measurability properties in uncountable time indices. A stopping time \tau: \Omega \to [0, \infty] with respect to the filtration (\mathcal{F}_t)_{t \geq 0} is formally defined as a measurable random variable such that, for every t \geq 0, the event \{\tau \leq t\} belongs to \mathcal{F}_t. This condition implies that the decision to stop by time t can be determined using only the information available up to t. Unlike in the discrete-time setting, where the events \{\tau \leq t\} and \{\tau < t+1\} coincide, in continuous time the event \{\tau < t\} generally does not belong to \mathcal{F}_t, although \{\tau \leq t\} does; this distinction arises due to the density of the time index and requires careful handling in proofs of measurability. Stopping times in continuous time are intimately related to the debut (or first entrance time) of progressive sets. A process A = (A_t)_{t \geq 0} with values in {0,1} is progressive with respect to (\mathcal{F}_t) if, for each t \geq 0, the map (s, \omega) \mapsto 1_{A_s}(\omega) from [0,t] \times \Omega to \mathbb{R} is measurable with respect to the product σ-algebra \mathcal{B}([0,t]) \otimes \mathcal{F}_t. The debut of such a set is defined as D(A)(\omega) = \inf\{t \geq 0 : \omega \in A_t\} (with the convention \inf \emptyset = \infty), and the debut theorem states that D(A) is a stopping time whenever A is progressive. Conversely, every stopping time \tau is the debut of the progressive set \{ (\tau \leq t) \}_{t \geq 0}. This characterization highlights why progressive measurability is essential for defining hitting times or first passage times as stopping times in continuous time. Many fundamental results on stopping times, such as those involving optional projections or martingale properties, assume that \tau is almost surely finite, meaning P(\tau < \infty) = 1. Without this, \tau may take the value \infty on a set of positive probability, complicating compositions with processes. Moreover, right-continuity of the filtration is necessary to guarantee that debuts of progressive sets are indeed stopping times; without it, measurability can fail for certain pathological sets. For instance, in non-right-continuous filtrations, the event \{\tau \leq t\} might not capture all relevant information at t, leading to inconsistencies in stochastic calculus applications.

Adapted Process Formulation

In the framework of stochastic processes, a process X = (X_t)_{t \geq 0} is adapted to a filtration (\mathcal{F}_t)_{t \geq 0} if, for each t \geq 0, the random variable X_t is \mathcal{F}_t-measurable. This adaptation ensures that the information revealed by the process up to time t is contained within the sigma-algebra \mathcal{F}_t, reflecting the progressive revelation of uncertainty in the underlying probability space. Stopping times can be reformulated in terms of adapted processes as the first instant at which the process satisfies a specified condition relative to the filtration. For an adapted process X and a Borel set C in the state space, a prototypical example is the first exit time \tau = \inf\{t \geq 0 : X_t \notin C\}, where \tau inherits the stopping time property from the adaptedness of X and the measurability of C. This formulation underscores the decision-making aspect of stopping times, as the exit event \{X_t \notin C\} is \mathcal{F}_t-measurable for each t, allowing decisions to be based solely on information available up to that time. Associated with a stopping time \tau, the stopped process X^\tau = (X^\tau_t)_{t \geq 0} is defined by X^\tau_t = X_{t \wedge \tau}, where a \wedge b = \min(a, b). This construction preserves adaptedness to the filtration (\mathcal{F}_t), as the minimum t \wedge \tau ensures that X^\tau_t depends only on information up to time t. The stopped process effectively freezes the evolution of X at \tau, facilitating analysis of behavior before and after the random stopping event without introducing extraneous information. A key structure induced by \tau is the sigma-algebra \mathcal{F}_\tau = \{A \in \mathcal{F}_\infty : A \cap \{\tau \leq t\} \in \mathcal{F}_t \ \forall t \geq 0\}, where \mathcal{F}_\infty = \bigvee_{t \geq 0} \mathcal{F}_t. This sigma-algebra captures the events observable precisely at the random time \tau, serving as the appropriate refinement of the terminal sigma-algebra \mathcal{F}_\infty for conditioning at irregular times. It enables the well-defined conditional expectation \mathbb{E}[Z \mid \mathcal{F}_\tau] for any integrable random variable Z that is \mathcal{F}_\infty-measurable, which is crucial for theorems involving expectations halted at stopping times. The family of sigma-algebras \{\mathcal{F}_\tau\} exhibits monotonicity: if \tau_1 \leq \tau_2 almost surely, then \mathcal{F}_{\tau_1} \subseteq \mathcal{F}_{\tau_2}. This increasing property allows for the construction of "stopped" filtrations, where the information structure respects the ordering of stopping times and supports sequential decision frameworks in stochastic models.

Properties

Basic Closure Properties

The set of stopping times is closed under various operations, ensuring flexibility in constructing new stopping times from existing ones. Constant times c \geq 0 (deterministic) are stopping times, as \{c = t\} \in \mathcal{F}_t for t \geq c and empty otherwise. If \tau and \sigma are stopping times, then their minimum \tau \wedge \sigma = \min(\tau, \sigma) and maximum \tau \vee \sigma = \max(\tau, \sigma) are also stopping times. Similarly, \tau \wedge n for fixed n is a stopping time. For a sequence of stopping times (\tau_k)_{k \geq 1}, the pointwise infimum \inf_k \tau_k and supremum \sup_k \tau_k are stopping times, provided the filtration is right-continuous in continuous time. Additionally, if S \leq T almost surely are stopping times, then the generated \sigma-algebra \mathcal{F}_S \subseteq \mathcal{F}_T, maintaining the adapted, non-anticipating structure. These closure properties allow composition and limits of stopping times, foundational for modeling complex decision processes in stochastic settings.

Localization

In stochastic processes, localization extends theorems or properties that hold for bounded stopping times to unbounded ones. For an unbounded stopping time \tau, a localizing sequence of stopping times (\tau_n)_{n \geq 1} is constructed such that \tau_n \uparrow \tau (or \infty) almost surely, typically defined as \tau_n = \tau \wedge n. This sequence bounds \tau progressively, controlling potential issues like infinite expectations or divergence. A property holds locally for \tau if it holds for each bounded \tau \wedge \tau_n almost surely. If a theorem (e.g., optional sampling) applies to bounded stopping times with respect to a right-continuous filtration, it extends to unbounded \tau via localization, under conditions like the filtration's completeness or uniform integrability of the underlying process. This propagates results such as martingale expectations at \tau without global boundedness. The importance of localization for stopping times lies in handling unbounded cases rigorously, avoiding failures in convergence or integration. For example, in the optional stopping theorem for martingales, bounded approximations \tau \wedge n preserve E[M_{\tau \wedge n}] = E[M_0], and under uniform integrability, the limit E[M_\tau] = E[M_0] holds as n \to \infty. This technique is crucial in stochastic calculus for unbounded domains.

Boundedness and Unboundedness

A stopping time \tau is bounded if there exists a finite constant M < \infty such that \tau \leq M almost surely; otherwise, it is unbounded. For a bounded stopping time \tau, the expectation E[\tau] is automatically finite, since \tau takes values in a finite interval with probability 1. Additionally, the event \{\tau \leq t\} belongs to the filtration \mathcal{F}_t for all t \geq M, which simplifies the measurability conditions inherent to the definition of stopping times. This boundedness ensures that optional sampling theorems apply directly without further restrictions; for instance, if X is a martingale, then E[X_\tau] = E[X_0]. Unbounded stopping times \tau may have E[\tau] = \infty, even if \tau < \infty almost surely (e.g., due to heavy-tailed distributions). However, in many applications, it is assumed that P(\tau < \infty) = 1, meaning \tau is almost surely finite, which ensures the stopped process X_{\tau \wedge t} is well-defined for all t > 0. Almost sure finiteness is a condition for extending results like optional sampling to unbounded cases, often requiring supplementary assumptions such as of the martingale or bounded increments. A fundamental convergence property holds for sequences of bounded stopping times: if \tau_n is an increasing sequence of bounded stopping times converging to \tau, and the filtration is right-continuous, then the associated \sigma-algebras satisfy \mathcal{F}_{\tau_n} \uparrow \mathcal{F}_\tau. This monotonicity of filtrations facilitates the analysis of limits in stochastic processes. Bounded stopping times play a central role in theorems where uniform integrability is readily satisfied, allowing straightforward application of martingale convergence results. For unbounded stopping times, more advanced tools are necessary, such as Doob's upcrossing inequality, which bounds the expected number of upcrossings of an interval by a supermartingale and enables almost sure convergence under L^1-boundedness, or localization techniques to reduce to bounded approximations.

Examples

[Category header - no content]

Basic Stochastic Process Examples

Basic examples illustrate stopping times in simple stochastic processes, such as a symmetric on integers where \tau = \min\{n \geq 0: S_n = a\} for a > 0 stops upon reaching level a. This \tau qualifies as a stopping time because \{\tau = n\} depends only on the path up to n, verifiable from the filtration \mathcal{F}_n = \sigma(S_0, \dots, S_n). Such examples highlight how stopping rules adapt to observed outcomes without foresight. In Poisson processes, the first event time \tau = \inf\{t > 0: N_t \geq 1\} serves as a stopping time, with \{\tau \leq t\} = \{N_t \geq 1\} \in \mathcal{F}_t. This demonstrates exponential waiting times as fundamental cases, where the stopping decision aligns with the natural filtration of jumps. Discrete analogs include , stopping at or target fortune. These examples underscore accessibility: the event of stopping is \mathcal{F}_t-measurable, enabling integration with martingale theory for expectation preservation. High-impact contributions, like Doob's foundational definitions, rely on such constructions to build broader theory.

Hitting Times

Hitting times, defined as \tau_A = \inf\{t \geq 0: X_t \in A\} for a set A and process X, are canonical stopping times when A is closed, as \{\tau_A \leq t\} = \bigcup_{s \leq t} \{X_s \in A\} \in \mathcal{F}_t. For , the first hitting time to a level b > 0 follows a , illustrating continuous-path properties. In Markov chains, hitting times to absorbing states model absorption probabilities, solvable via first-step analysis: h_i = \mathbb{P}(\tau_A < \infty | X_0 = i) satisfies linear equations from transitions. Exit times from intervals, like \tau = \inf\{n: S_n \notin (0,a)\} in random walks, exemplify bounded cases if the process is recurrent. Hitting times drive applications in diffusion processes, where Wald's identities compute expectations under optional stopping. Seminal papers by Lévy on Brownian hitting times establish scaling laws, such as \mathbb{E}[\tau_b] = \infty for one-dimensional Brownian motion hitting b \neq 0. These examples distinguish hitting from non-stopping times like last exits, which require future information.

Applications

[Category header - no content]

Clinical Trials

In clinical trials, stopping times enable sequential , halting when or futility thresholds are crossed, as in group sequential designs by Pocock or O'Brien-Fleming boundaries. For example, the Lan-DeMets alpha-spending function approximates these, allowing early termination if interim p-values fall below adjusted limits, balancing type I error. Futility stopping occurs when conditional power drops below 20%, computed from interim , preventing resource waste on ineffective treatments. Safety stopping rules trigger if adverse events exceed predefined rates, often using exact tests. trial exemplifies stringent thresholds (p < 0.0001 early on) for benefit stopping, ensuring robust evidence. These applications draw from martingale theory, where cumulative statistics form martingales stopped at boundaries, preserving overall significance levels. Influential guidelines from ICH harmonized principles mandate such rules in licensure trials, prioritizing and efficiency. Early stopping for benefit risks overestimation, mitigated by conservative boundaries in high-impact studies.

Martingale Theory

Stopping times are central to martingale theory via the , stating that for a martingale M_t and bounded stopping time \tau, \mathbb{E}[M_\tau] = \mathbb{E}[M_0], preserving the martingale property at random horizons. Extensions to unbounded times require , as in Doob's theorem for submartingales. In discrete time, the theorem applies to fair games, showing no gain from stopping strategies, exemplified by where fortune at ruin equals initial stake in . Continuous-time versions handle diffusions, with stopped processes \tilde{M}_t = M_{t \wedge \tau} remaining martingales. Applications include Wald's , stopping upon crossing log-likelihood boundaries, yielding optimal error rates. Seminal contributions by Doob (1953) formalize these, influencing and via risk-neutral pricing. The theorem's conditions—bounded or finite moments—ensure applicability without from premature stopping.

Types

Stopping times are classified into accessible and totally inaccessible types, with predictable stopping times forming an important subclass of accessible ones. Accessible stopping times are those that can be announced by a sequence of predictable stopping times converging to them almost surely.

Predictable Stopping Times

Predictable stopping times are those announced by an increasing sequence of stopping times \tau_n \uparrow \tau almost surely, allowing prediction just before occurrence: \{\tau = t\} \in \mathcal{F}_{t-} for discrete times, or via left-continuous adapted processes in continuous settings. Examples include deterministic times t_0 or first jumps in predictable compensators. In filtration theory, predictability implies the stochastic interval ([\tau, \infty)) is predictable, enabling decomposition of integrals over predictable sets. For predictable finite variation processes, jumps occur only at predictable times, contrasting with inaccessible ones. Doob-Meyer decomposition relies on this for compensators of submartingales. These times facilitate optional sampling without accessibility issues, as in announced stopping rules like fixed horizons. High-impact work by Dellacherie and Meyer classifies them within progressive enlargements, essential for modeling where default is predictable via intensity processes.

Totally Inaccessible Stopping Times

Totally inaccessible stopping times defy prediction, satisfying \mathbb{P}(\tau = \sigma < \infty) = 0 for any predictable \sigma, with no announcing sequence converging without equality. The first jump of a process exemplifies this, as jumps arrive unexpectedly relative to the natural . In continuous-time, they decompose general stopping times via the accessible part (predictable) and totally inaccessible part, as in the Dellacherie-Meyer theorem, which decomposes the graph of any stopping time \tau into disjoint accessible and totally inaccessible parts, equivalently partitioning \{\tau < \infty\} such that \tau coincides with an accessible stopping time on the accessible part and with a totally inaccessible one on the inaccessible part. Jumps at such times drive martingale discontinuities, with compensators zero on predictable parts. Applications include honest times in enlargement of filtrations, where immersion holds if inaccessibility is total. Seminal results by Jeulin and Yor link them to last passage times, impacting volatility models in where sudden shocks are inaccessible. The dichotomy ensures exhaustive classification, with probability measures singular on accessible vs. inaccessible sets.