Stopping time is the mathematical language of deciding when to stop observing a random process without using information from the future. A gambler quits when her capital reaches a target. A clinical trial stops when the evidence crosses a safety boundary. A [Markov chain](/page/Markov%20Chain) is observed until it first enters an absorbing state. A Brownian particle is watched until it first hits a barrier. In each case, the stopping rule is allowed to look at what has happened so far. It is not allowed to look ahead. This information principle is the central idea. At time $n$, the event that the rule has already stopped must be knowable from the information available by time $n$. The formal setting is a filtered [probability space](/page/Probability%20Space). Here $\mathcal{F}_n$ records all events whose truth can be decided after observing the process up to time $n$. The filtration $(\mathcal{F}_n)_{n \geq 0}$ is increasing, so information is never forgotten. This chapter works primarily in discrete time. Continuous time uses the same idea, with $t \geq 0$ replacing integer times $n \geq 0$ and with technical regularity assumptions often imposed on the filtration. ## Definition This page uses the discrete-time convention that processes are indexed by $0,1,2,\ldots$. This differs from the global convention that $\mathbb{N}$ starts at $1$, so integer time will be written explicitly as $n \geq 0$. A [random variable](/page/Random%20Variable) $\tau$ taking values in $\{0,1,2,\ldots\} \cup \{\infty\}$ may name a time. It becomes a stopping time only when the decision to stop respects the filtration. The definition below encodes the information principle directly. At time $n$, it must already be possible to decide whether stopping has occurred by time $n$, because a rule that needs later observations is not a legitimate rule for stopping at time $n$. [definition: Stopping Time] Let $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \geq 0},\mathbb{P})$ be a filtered probability space. A map $\tau:\Omega \to \{0,1,2,\ldots\} \cup \{\infty\}$ is a stopping time with respect to $(\mathcal{F}_n)$ if \begin{align*} \{\omega \in \Omega : \tau(\omega) \leq n\} \in \mathcal{F}_n \end{align*} for every integer $n \geq 0$. [/definition] ## Information and Adaptedness The definition is not only about the random time $\tau$; it is also about the calendar of information against which $\tau$ is judged. Before we can compare stopping rules across examples, we need a single object that records the probability space together with the increasing family of observations. The same formula for $\tau$ can be legitimate under a rich information flow and illegitimate under a smaller one, so the filtration must be part of the ambient structure rather than an afterthought. [definition: Filtered Probability Space] A discrete-time filtered probability space is a tuple $(\Omega, \mathcal{F}, (\mathcal{F}_n)_{n \geq 0}, \mathbb{P})$ such that $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space, each $\mathcal{F}_n$ is a sub-$\sigma$-algebra of $\mathcal{F}$, and $\mathcal{F}_n \subset \mathcal{F}_{n+1}$ for every integer $n \geq 0$. [/definition] The simplest stopping rules do not depend on the data at all. They are useful as boundary cases and as deterministic horizons in optional sampling, because they show that deterministic observation windows are included in the stopping-time framework. [example: Deterministic Time] Fix an integer $m \geq 0$ and define $\tau(\omega)=m$ for every $\omega \in \Omega$. To check the stopping-time condition, fix an integer $n \geq 0$ and compute the event \begin{align*} \{\tau \leq n\}=\{\omega \in \Omega : m \leq n\}. \end{align*} If $n<m$, then the inequality $m \leq n$ is false for every $\omega$, so \begin{align*} \{\tau \leq n\}=\varnothing. \end{align*} If $n \geq m$, then the inequality $m \leq n$ is true for every $\omega$, so \begin{align*} \{\tau \leq n\}=\Omega. \end{align*} Since every $\sigma$-algebra contains both $\varnothing$ and $\Omega$, in both cases $\{\tau \leq n\}\in \mathcal{F}_n$. Thus $\tau$ is a stopping time, and the example shows that a deterministic calendar time is automatically compatible with any filtration. [/example] The condition $\{\tau \leq n\} \in \mathcal{F}_n$ is often the most natural one. In discrete time, it is equivalent to asking that the exact event of stopping at time $n$ be observable at time $n$, and this version is useful when constructing stopping times from first hitting rules. [quotetheorem:9935] This characterization is more than a change of notation. Its necessity says that a legitimate rule must reveal, by time $n$, whether the rule stopped exactly at time $n$; its sufficiency says that these exact-time observations recover the usual stopping-time event by the finite union \begin{align*} \{\tau \leq n\}=\bigcup_{k=0}^n \{\tau=k\}. \end{align*} That finite-union step is why the statement is especially useful in discrete time, and also why it should not be copied verbatim into continuous time without extra hypotheses. In practice, the result is the bridge from definitions to examples: first hitting times, threshold-crossing rules, and sequential tests can often be checked by proving that the event of first stopping at a fixed time depends only on observations available at that time. Stopping times are also called optional times. The word optional emphasizes that the time may depend on the evolving observation, but it does not mean that arbitrary future-dependent choices are allowed. The stopping decision must be adapted to the information flow. To connect stopping rules with observed data, we need a name for processes whose present value is actually visible at the present time. Without this condition, a first entrance rule could secretly depend on unobservable variables hidden inside the process. [definition: Adapted Process] Let $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \geq 0},\mathbb{P})$ be a filtered probability space and let $(E,\mathcal{E})$ be a measurable space. A process $X=(X_n)_{n \geq 0}$ with $X_n:\Omega \to E$ is adapted to $(\mathcal{F}_n)$ if $X_n$ is $\mathcal{F}_n$-$\mathcal{E}$ measurable for every integer $n \geq 0$. [/definition] This says that the value of $X_n$ is observable by time $n$. Adaptedness links stopping times to observed trajectories. ## Hitting Times ### Entrance Rules Many stopping times arise as first entrance times. The observer waits until a process enters a target set, and this model appears in ruin problems, queueing, Markov chains, finance, and sequential statistics. [definition: First Entrance Time] Let $(E,\mathcal{E})$ be a measurable space, let $X=(X_n)_{n \geq 0}$ be a process with $X_n:\Omega \to E$ for every integer $n \geq 0$, and let $A \in \mathcal{E}$. The first entrance time of $X$ into $A$ is the map $\tau_A:\Omega \to \{0,1,2,\ldots\}\cup\{\infty\}$ defined by \begin{align*} \tau_A(\omega)=\inf\{n \geq 0: X_n(\omega) \in A\}, \end{align*} with the convention $\inf \varnothing=\infty$. [/definition] The next result justifies the common practice of stopping when an observed process first reaches a measurable region. It is the basic mechanism behind hitting times for adapted processes, because first entrance can be detected by checking finitely many observations up to the present time. [quotetheorem:9936] This result separates two issues that are easy to confuse: the pathwise definition of the entrance time and the measurability needed for the time to be observable. The adaptedness hypothesis is essential; without it, the event that the process has entered $A$ by time $n$ need not be determined by $\mathcal{F}_n$. In the examples below, this theorem is the reason that hitting a boundary is a legitimate stopping rule rather than a rule that secretly looks into the future. ### Random Walk Boundaries First hitting times give concrete stopping rules in random walks. They express a rule based only on the present and past values of the process, so they are the prototype of a path-dependent but non-anticipative stopping rule. [example: First Hit of a Random Walk] Let $(S_n)_{n \geq 0}$ be a real-valued random walk adapted to $(\mathcal{F}_n)$, and fix $a \in \mathbb{R}$. Define \begin{align*} \tau_a=\inf\{k \geq 0: S_k \geq a\}. \end{align*} We show that $\tau_a$ is a stopping time. Fix an integer $n \geq 0$. By the definition of $\tau_a$, the event $\{\tau_a \leq n\}$ occurs exactly when some time $k$ with $0 \leq k \leq n$ satisfies $S_k \geq a$, so \begin{align*} \{\tau_a \leq n\}=\bigcup_{k=0}^{n}\{S_k \geq a\}. \end{align*} For each $k \leq n$, adaptedness gives $\{S_k \geq a\}\in\mathcal{F}_k$, because $\{S_k \geq a\}=S_k^{-1}([a,\infty))$ and $[a,\infty)$ is Borel measurable. Since the filtration is increasing and $k \leq n$, we have $\mathcal{F}_k \subset \mathcal{F}_n$, hence $\{S_k \geq a\}\in\mathcal{F}_n$ for every $k=0,\ldots,n$. A finite union of events in the $\sigma$-algebra $\mathcal{F}_n$ again belongs to $\mathcal{F}_n$, so \begin{align*} \{\tau_a \leq n\}\in\mathcal{F}_n. \end{align*} This holds for every integer $n \geq 0$, so $\tau_a$ is a stopping time: the rule stops the first time the observed random walk reaches the level $a$, using no information beyond the present time. [/example] Some random times look natural but are not stopping times. The test is whether the answer at time $n$ requires future information. The next example is a standard warning. [example: Last Success Before a Fixed Horizon] Let $N \geq 1$ and take the canonical sample space $\Omega=\{0,1\}^N$, with $Y_k(\omega)=\omega_k$. The filtration is $\mathcal{F}_0=\{\varnothing,\Omega\}$ and $\mathcal{F}_n=\sigma(Y_1,\ldots,Y_n)$ for $1 \leq n \leq N$. Define \begin{align*} \tau=\max\{k \leq N: Y_k=1\}, \end{align*} with the convention $\tau=0$ if $Y_1=\cdots=Y_N=0$. We test the stopping-time condition at time $N-1$. The event $\{\tau \leq N-1\}$ means that the last success occurs no later than $N-1$, which is equivalent to saying that there is no success at time $N$. Thus \begin{align*} \{\tau \leq N-1\}=\{\omega \in \Omega : Y_N(\omega)=0\}. \end{align*} This event is not generally in $\mathcal{F}_{N-1}=\sigma(Y_1,\ldots,Y_{N-1})$. To see this explicitly, let \begin{align*} \omega=(0,0,\ldots,0,0) \end{align*} and \begin{align*} \omega'=(0,0,\ldots,0,1). \end{align*} These two outcomes have the same first $N-1$ coordinates, so every event in $\sigma(Y_1,\ldots,Y_{N-1})$ must either contain both of them or contain neither of them. But $Y_N(\omega)=0$, so $\omega \in \{\tau \leq N-1\}$, while $Y_N(\omega')=1$, so $\omega' \notin \{\tau \leq N-1\}$. Therefore \begin{align*} \{\tau \leq N-1\}\notin \mathcal{F}_{N-1}. \end{align*} Hence $\tau$ is not a stopping time in this model. The obstruction is exactly chronological: at time $N-1$, one cannot know whether time $N$ will contain a later success. [/example] This distinction is not cosmetic. A theorem about stopping a [Martingale](/page/Martingale) at a stopping time may fail when the time is chosen using the future. The filtration is the safeguard against such anticipative choices. ## Stopping Rules and Sigma Algebras ### Stopped Information When a process is stopped at a random time, the information available at that random time also needs a formal description. The stopped $\sigma$-algebra collects events whose truth can be decided once the stopping time occurs, and this construction is central to conditioning at random times and to optional sampling. [definition: Stopped Sigma Algebra] Let $\tau$ be a stopping time on $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \geq 0},\mathbb{P})$. The stopped $\sigma$-algebra $\mathcal{F}_\tau$ is \begin{align*} \mathcal{F}_\tau = \{A \in \mathcal{F}: A \cap \{\tau \leq n\} \in \mathcal{F}_n \text{ for every integer } n \geq 0\}. \end{align*} [/definition] The stopped $\sigma$-algebra should be read as the information known at the random time $\tau$. To use it for conditioning and measurability arguments, we first need to know that the displayed collection is actually a $\sigma$-algebra rather than only suggestive notation. [quotetheorem:9937] The theorem is the point at which the notation $\mathcal{F}_\tau$ becomes usable as an object of probability theory. Since conditional expectations and measurability statements are only formed relative to a genuine $\sigma$-algebra, this closure result is not cosmetic: it is what allows one to speak rigorously about information available at the random time. The definition also depends on the stopping-time property; if $\tau$ could depend on future information, the sets used to test membership in $\mathcal{F}_\tau$ would no longer reflect what is knowable by time $n$. ### Stopped Processes A stopped process freezes a trajectory once the stopping rule has triggered. This converts a random-horizon observation into an ordinary adapted process indexed by deterministic times, which is why stopped processes are the main objects used in optional sampling arguments. [definition: Stopped Process] Let $(E,\mathcal{E})$ be a measurable space, let $X=(X_n)_{n \geq 0}$ be a process with $X_n:\Omega \to E$ for every integer $n \geq 0$, and let $\tau:\Omega \to \{0,1,2,\ldots\}\cup\{\infty\}$ be a stopping time. The stopped process $X^\tau=(X^\tau_n)_{n \geq 0}$ is the process with $X^\tau_n:\Omega \to E$ defined by \begin{align*} X^\tau_n(\omega) = X_{n \wedge \tau(\omega)}(\omega) \end{align*} for every integer $n \geq 0$ and every $\omega \in \Omega$, with the convention $n \wedge \infty=n$. [/definition] The stopped process is the path that follows $X$ until $\tau$ and then remains fixed. In gambling language, it is the capital process after the gambler has quit. In statistics, it is the accumulated statistic after a trial is stopped. In finance, it is the price process observed only until a trading rule exits. ## Martingales at Stopping Times Stopping times are especially powerful when paired with martingales. A [Martingale](/page/Martingale) models a fair game relative to a filtration. Optional sampling explains when stopping a fair game keeps it fair. The bounded version is the safest and most frequently used first form, because a deterministic upper bound reduces the random-time calculation to finitely many ordinary martingale identities. [quotetheorem:1153] The boundedness hypothesis matters. Without boundedness or additional integrability assumptions, optional sampling can fail. A strategy may wait for a rare favorable event and create a misleading appearance of profit. The theorem does not say that gambling systems beat fair games. It says that fair games remain fair under stopping rules that satisfy the required hypotheses. The proof belongs to martingale theory and relies on [Conditional Expectation](/page/Conditional%20Expectation). A gambler's stopping rule is admissible when it is based on observed wins and losses. It is inadmissible when it depends on later outcomes. [example: Gambling Until a Target] Let $(M_n)_{n \geq 0}$ be the capital process, adapted to the natural filtration $(\mathcal{F}_n)$, and fix constants $a<M_0<b$. Define \begin{align*} \tau=\inf\{n \geq 0: M_n \geq b \text{ or } M_n \leq a\}. \end{align*} We verify that this target-exit rule is a stopping time. For an integer $n \geq 0$, the event $\{\tau \leq n\}$ occurs exactly when the capital has crossed one of the two barriers at some observed time $k \leq n$, so \begin{align*} \{\tau \leq n\}=\bigcup_{k=0}^{n}\bigl(\{M_k \geq b\}\cup\{M_k \leq a\}\bigr). \end{align*} For each $k \leq n$, adaptedness gives that $M_k$ is $\mathcal{F}_k$-measurable. Since $[b,\infty)$ and $(-\infty,a]$ are Borel subsets of $\mathbb{R}$, this gives \begin{align*} \{M_k \geq b\}=M_k^{-1}([b,\infty))\in\mathcal{F}_k \end{align*} and \begin{align*} \{M_k \leq a\}=M_k^{-1}((-\infty,a])\in\mathcal{F}_k. \end{align*} Because the filtration is increasing and $k \leq n$, we have $\mathcal{F}_k\subset\mathcal{F}_n$, hence \begin{align*} \{M_k \geq b\}\cup\{M_k \leq a\}\in\mathcal{F}_n \end{align*} for every $k=0,\ldots,n$. The displayed union is finite, so it also belongs to $\mathcal{F}_n$. Therefore \begin{align*} \{\tau \leq n\}\in\mathcal{F}_n \end{align*} for every integer $n \geq 0$, and $\tau$ is a stopping time. The rule is legitimate because it only asks whether the already observed capital has reached either boundary. Once boundedness or the needed integrability hypotheses are added, optional sampling applies to this stopped fair game. [/example] ## Bounded and Integrable Stopping Times The class of stopping times has useful closure properties. These properties let complicated rules be assembled from simpler ones. The minimum of two stopping times means stopping when the first of two rules triggers. The maximum means waiting until both rules have triggered, and both operations preserve the rule that the stopping decision must be knowable from current information. [quotetheorem:9938] Finite combinations behave better than infinite combinations because the filtration condition is tested at fixed finite times. Sums are less immediate than minima and maxima: by time $n$, one must decide whether the first rule has already stopped at some earlier time and whether the remaining waiting time has elapsed using only the information available up to $n$. The point of the discrete-time result is to show that this bookkeeping still respects the filtration, so two successive waiting rules can be combined without looking into the future. [quotetheorem:9939] Sum closure is useful because many stopping rules are naturally staged: wait until one observable event occurs, then start a second rule using the same evolving information. The theorem guarantees that this successive construction is still a stopping time when both stages are adapted to the same filtration. Its limitation is equally important: it does not justify restarting with extra future information or with a different information flow unless the relevant measurability conditions are checked separately. The next issue is not whether a random time is observable, but whether it can be controlled analytically. Optional sampling and stopped-process arguments become much easier when the random horizon cannot run forever. Without a deterministic cap, estimates may have to control behavior at arbitrarily late times, where convergence and integrability issues can enter. The boundedness condition rules this out by requiring the random time to lie below a fixed non-random horizon almost surely. [definition: Bounded Stopping Time] Let $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \geq 0},\mathbb{P})$ be a filtered probability space. A stopping time $\tau:\Omega \to \{0,1,2,\ldots\}\cup\{\infty\}$ with respect to $(\mathcal{F}_n)$ is bounded if there exists an integer $N \geq 0$ such that \begin{align*} \mathbb{P}(\tau \leq N)=1. \end{align*} [/definition] Many applications start with bounded stopping times and then pass to limits. This limiting step is delicate, and it usually requires [uniform integrability](/page/Uniform%20Integrability), domination, or convergence hypotheses to keep the limiting random time from smuggling in uncontrolled behavior. [definition: Integrable Stopping Time] Let $(\Omega,\mathcal{F},(\mathcal{F}_n)_{n \geq 0},\mathbb{P})$ be a filtered probability space. A stopping time $\tau:\Omega \to \{0,1,2,\ldots\}\cup\{\infty\}$ with respect to $(\mathcal{F}_n)$ is integrable if \begin{align*} \mathbb{E}[\tau] < \infty. \end{align*} [/definition] Integrability forces $\mathbb{P}(\tau<\infty)=1$, since a positive probability of $\tau=\infty$ would make the expectation infinite. Integrability of $\tau$ alone is not a universal substitute for boundedness. It can be enough in special theorems involving bounded increments or stronger martingale assumptions. The hypotheses must be matched to the conclusion. ## Sequential Testing Stopping times formalize data-dependent sample sizes. In sequential testing, observations arrive one at a time, and the experiment stops when enough evidence has accumulated. The rule is valid only if the stopping decision at time $n$ uses the first $n$ observations. [example: Sequential Likelihood Boundary] Let $Z_1,Z_2,\ldots$ be observations and let $\mathcal{F}_n=\sigma(Z_1,\ldots,Z_n)$. Suppose that each likelihood ratio $L_n$ is $\mathcal{F}_n$-measurable, as it is computed from the observations $Z_1,\ldots,Z_n$. For constants $0<a<b$, define \begin{align*} \tau=\inf\{n \geq 1: L_n \leq a \text{ or } L_n \geq b\}. \end{align*} We verify the stopping-time condition. At time $0$, the set in the infimum cannot contain any index $n \leq 0$, so \begin{align*} \{\tau \leq 0\}=\varnothing. \end{align*} Since $\varnothing \in \mathcal{F}_0$, the condition holds for $n=0$. Now fix an integer $n \geq 1$. The event $\{\tau \leq n\}$ occurs exactly when at least one of the tests at times $1,\ldots,n$ has crossed a boundary, so \begin{align*} \{\tau \leq n\}=\bigcup_{k=1}^{n}\bigl(\{L_k \leq a\}\cup\{L_k \geq b\}\bigr). \end{align*} For each $k \leq n$, measurability of $L_k$ with respect to $\mathcal{F}_k$ gives \begin{align*} \{L_k \leq a\}=L_k^{-1}((-\infty,a])\in\mathcal{F}_k \end{align*} because $(-\infty,a]$ is a Borel subset of $\mathbb{R}$. Similarly, \begin{align*} \{L_k \geq b\}=L_k^{-1}([b,\infty))\in\mathcal{F}_k. \end{align*} Since the filtration is increasing and $k \leq n$, we have $\mathcal{F}_k\subset\mathcal{F}_n$, hence \begin{align*} \{L_k \leq a\}\cup\{L_k \geq b\}\in\mathcal{F}_n. \end{align*} The union over $k=1,\ldots,n$ is finite, so it also belongs to $\mathcal{F}_n$. Therefore \begin{align*} \{\tau \leq n\}\in\mathcal{F}_n \end{align*} for every integer $n \geq 0$, and $\tau$ is a stopping time. The sequential test is legitimate because the decision to stop at time $n$ depends only on likelihood ratios already computed from the first $n$ observations. [/example] This example explains why stopped experiments can be analyzed rigorously. The random sample size is not a defect by itself. The crucial point is that the stopping criterion is adapted. If a trial stops because an unobserved future dataset would have been unfavorable, the rule is not a stopping time. ## Markov Chain Hitting Times For a Markov chain, hitting times are among the most important random times. They describe first visits to states, return times, absorption times, and entrance into target sets. Let $(X_n)$ be a Markov chain on a countable state space $E$. The natural filtration is $\mathcal{F}_n=\sigma(X_0,\ldots,X_n)$. For $A \subset E$, the hitting time \begin{align*} T_A=\inf\{n \geq 0: X_n \in A\} \end{align*} is a stopping time. This fact is part of the bridge between stopping times and the [strong Markov property](/theorems/2208). At a stopping time, a Markov chain can often be restarted from its random state $X_\tau$. The restart statement requires a theorem, not just notation, because the stopping-time condition is what prevents the restart time from depending on future states. [quotetheorem:2208] The strong Markov property is stronger than the ordinary Markov property because the restart time is random rather than deterministic. The stopping-time hypothesis is what makes this possible: it ensures that the decision to restart has been made using only information already revealed, so the future evolution can be compared with a fresh process started from the random state at $\tau$. The statement does not apply to arbitrary random times, and it should be read as a theorem about compatible random times and Markov processes rather than as a general independence principle. ## Brownian Hitting Times In continuous time, a stopping time $\tau$ satisfies $\{\tau \leq t\}\in \mathcal{F}_t$ for every $t \geq 0$. The same information principle applies. The technical landscape is richer because time is uncountable. Filtration regularity conditions, such as right-continuity and completion, often enter. For [Brownian Motion](/page/Brownian%20Motion), hitting times of barriers are central examples, but the exact statement depends on the path regularity and filtration hypotheses. With continuous paths and the usual right-continuous completed filtration, the standard one-dimensional level-hitting times are stopping times. If $(B_t)_{t \geq 0}$ is Brownian motion with this usual filtration, the first time it reaches level $a$ is \begin{align*} T_a=\inf\{t \geq 0: B_t=a\}. \end{align*} Under the standard continuous-path setup, $T_a$ is a stopping time. For fixed $t$, continuity of the path lets the event $\{T_a \leq t\}$ be recovered from the path on $[0,t]$, and the usual right-continuous completed filtration supplies the measurability needed to place that event in $\mathcal{F}_t$. Brownian hitting times are used in barrier options, diffusion theory, potential theory, and [stochastic calculus](/page/Stochastic%20Calculus). The stopping-time concept is also essential for local martingales. A local martingale is often defined through a sequence of stopping times that make the process a true martingale on bounded windows. ## The Information Principle A stopping time is a test of temporal honesty. For every $n$, the rule must answer the question “have we stopped by now?” using only $\mathcal{F}_n$. It may use the whole past. It may use the current observation. It may not use later observations. This principle explains why first hitting times are stopping times. It also explains why last hitting times before a fixed horizon often are not. A first occurrence can be recognized when it happens. A last occurrence is usually confirmed only after seeing that no later occurrence happens. The condition $\{\tau \leq n\}\in\mathcal{F}_n$ is therefore stronger than measurability of $\tau$ as a random variable. A random time can be measurable with respect to $\mathcal{F}$ while still failing to be a stopping time. The full $\sigma$-algebra $\mathcal{F}$ may contain information from the entire experiment. The filtration remembers when information becomes available. ## Stopped Values If $X$ is adapted and $\tau$ is a stopping time, the stopped value $X_\tau$ is a random variable defined on $\{\tau<\infty\}$. In discrete time it can be written through the partition $\{\tau=n\}$. Formally, \begin{align*} X_\tau = \sum_{n=0}^{\infty} X_n \mathbb{1}_{\{\tau=n\}} \end{align*} on the event $\{\tau<\infty\}$, whenever the expression is meaningful in a discrete-time model indexed by $0,1,2,\ldots$. This representation shows how stopped values combine observations at deterministic times. Measurability of $X_\tau$ with respect to $\mathcal{F}_\tau$ is a basic compatibility result, because the value observed at a legitimate random time should be measurable with respect to the information available at that time. [quotetheorem:9940] This theorem is the measurability guarantee that makes stopped values usable in later arguments. It says that evaluating an adapted process at a stopping time does not create extra information: on the event $\{\tau=n\}$, the value is the already observable $X_n$, and the event $\{\tau=n\}$ is itself decided by time $n$. This is the mechanism behind optional sampling, conditioning on stopped information, and localization by bounded stopping times. The hypotheses matter: if $X$ is not adapted, then $X_n$ may already contain future information, and if $\tau$ is not a stopping time, the choice of which $X_n$ to reveal may depend on information unavailable when the stop is declared. ## Beyond and Connected Topics Stopping times are the first layer of random-time analysis. More refined notions include predictable times, optional processes, progressive measurability, and debut theorems. These concepts become important in continuous-time stochastic processes, especially alongside [Brownian Motion](/page/Brownian%20Motion) and [Conditional Expectation](/page/Conditional%20Expectation). A predictable time can be announced just before it occurs. A totally inaccessible time cannot be forecast in that way. In jump processes, this distinction separates scheduled jumps from surprise jumps. Stopping times also support localization. A process may be too wild globally but well behaved before each time in an increasing sequence of stopping times. Such sequences are used to define local martingales, semimartingales, and stochastic integrals. The core idea remains unchanged. A legitimate random time must be tied to the information actually available. The condition is not a technical ornament. It is what allows conditional expectation, martingale fairness, Markov restart, and sequential inference to make mathematical sense at random times. The stopping-time framework is therefore both a modeling discipline and a theorem-making device. It lets random horizons be treated with the same rigor as deterministic horizons while preserving the chronology of information. ## References Androma, [Cambridge III Advanced Probability](/page/Cambridge%20III%20Advanced%20Probability). Androma, [Cambridge III Stochastic Calculus and Applications](/page/Cambridge%20III%20Stochastic%20Calculus%20and%20Applications). Androma, [Cambridge IB Probability and Measure](/page/Cambridge%20IB%20Probability%20and%20Measure). Androma, [Cambridge IA Probability](/page/Cambridge%20IA%20Probability). Androma, [Cambridge IB Markov Chains](/page/Cambridge%20IB%20Markov%20Chains). Androma, [Martingale](/page/Martingale). Androma, [Conditional Expectation](/page/Conditional%20Expectation). Androma, [Brownian Motion](/page/Brownian%20Motion). David Williams, *Probability with Martingales* (1991). Rick Durrett, *Probability: Theory and Examples* (2019). Daniel Revuz and Marc Yor, *Continuous Martingales and Brownian Motion* (1999).