A filtration specifies what is known at each time. Adaptedness prevents a process from using future information; predictability strengthens this to information available immediately before an increment; and a stopping time is a random decision time detectable without looking ahead. These distinctions are the measurable foundation of martingale and stochastic-integration theory.
Information flows
Definition. On \((\Omega,\mathcal F,P)\), a filtration indexed by \(T=\mathbb N_0\) or \(\mathbb R_+\) is a family \((\mathcal F_t)_{t\in T}\) such that \[ \mathcal F_s\subseteq\mathcal F_t\subseteq\mathcal F \qquad(s\le t). \] A stochastic process \(X=(X_t)\) is adapted if \(X_t\) is \(\mathcal F_t\)-measurable for each \(t\).
Definition. The natural filtration of a process \(X\) is \[ \mathcal F_t^X=\sigma(X_s:0\le s\le t). \] It is the smallest filtration to which \(X\) is adapted. A filtration in continuous time satisfies the usual conditions if it is complete and right-continuous: \[ \mathcal F_t=\mathcal F_{t+}:=\bigcap_{s>t}\mathcal F_s. \]
Completion places every subset of a null event in \(\mathcal F_0\). Right continuity rules out information appearing immediately after \(t\) but at no strictly positive delay. Both conditions remove avoidable technical pathologies in continuous-time stopping arguments.
Definition (discrete predictability). A sequence \(H_n\), \(n\ge1\), is predictable if \(H_n\) is \(\mathcal F_{n-1}\)-measurable. Thus the coefficient multiplying the increment from \(n-1\) to \(n\) must be chosen before that increment is observed.
Stopping times
Definition. A map \(\tau:\Omega\to T\cup\{\infty\}\) is a stopping time if \[ \{\tau\le t\}\in\mathcal F_t \qquad\text{for every }t\in T. \] A random time need not be a stopping time. For example, the last time a random walk visits zero before a fixed horizon generally depends on future observations.
Proposition (discrete criteria). For \(T=\mathbb N_0\), the following are equivalent: \[ \{\tau\le n\}\in\mathcal F_n,\qquad \{\tau=n\}\in\mathcal F_n,\qquad \{\tau>n\}\in\mathcal F_n \] for every \(n\).
Proof. Complements give the equivalence of the first and third conditions. Also \[ \{\tau=n\}=\{\tau\le n\}\setminus\{\tau\le n-1\}\in\mathcal F_n. \] Conversely, \(\{\tau\le n\}=\bigcup_{k=0}^n\{\tau=k\}\in\mathcal F_n\), since \(\mathcal F_k\subseteq\mathcal F_n\).
Proposition (closure operations). If \(\sigma,\tau\) are stopping times, then \(\sigma\wedge\tau\) and \(\sigma\vee\tau\) are stopping times. If \(\tau_n\) are stopping times, then \(\inf_n\tau_n\) is a stopping time. Under right continuity of the filtration, \(\sup_n\tau_n\) is also a stopping time whenever the supremum is the increasing limit of \(\tau_n\).
Proof. The identities \[ \{\sigma\wedge\tau\le t\} =\{\sigma\le t\}\cup\{\tau\le t\}, \quad \{\sigma\vee\tau\le t\} =\{\sigma\le t\}\cap\{\tau\le t\} \] prove the first claim, and a countable union handles \(\inf_n\tau_n\). If \(\tau_n\uparrow\tau\), then \[ \{\tau\le t\} =\bigcap_{m=1}^{\infty}\bigcup_{N=1}^{\infty} \bigcap_{n\ge N}\{\tau_n<t+1/m\}\in \bigcap_{m}\mathcal F_{t+1/m}=\mathcal F_t. \]
Hitting and entrance times
Theorem. Let \(X\) be an adapted continuous process with values in a metric space, and let \(F\) be closed. Then \[ \tau_F=\inf\{t\ge0:X_t\in F\} \] is a stopping time. If \(G\) is open and \(X\) is adapted with right-continuous paths, then the entrance time \(\tau_G=\inf\{t\ge0:X_t\in G\}\) is a stopping time.
Proof. For closed \(F\) and continuous paths, \[ \{\tau_F\le t\} = \left\{\inf_{0\le s\le t}d(X_s,F)=0\right\} = \left\{\inf_{s\in\mathbb Q\cap[0,t]}d(X_s,F)=0\right\}. \] The final event belongs to \(\mathcal F_t\) because it is determined by countably many adapted coordinates. For open \(G\), \[ \{\tau_G<t\} =\bigcup_{s\in\mathbb Q\cap[0,t)}\{X_s\in G\} \in\mathcal F_t, \] where right continuity ensures that an entrance before \(t\) is detected at a nearby rational time. Under a right-continuous filtration, \(\{\tau_G\le t\}=\bigcap_m\{\tau_G<t+1/m\}\in\mathcal F_t\).
Example. For an adapted random walk \(S_n\), the first passage time \[ \tau_a=\inf\{n\ge0:S_n\ge a\} \] is a stopping time because \(\{\tau_a\le n\}=\{\max_{k\le n}S_k\ge a\}\in\mathcal F_n\). By contrast, \(\sup\{k\le n:S_k=0\}\) is usually not a stopping time: deciding that a visit is the last one requires observing the future through \(n\).
Information at a stopping time
Definition. For a stopping time \(\tau\), define \[ \mathcal F_\tau = \{A\in\mathcal F: A\cap\{\tau\le t\}\in\mathcal F_t \text{ for every }t\}. \] This is the sigma-field of events decidable by time \(\tau\).
Proposition. The class \(\mathcal F_\tau\) is a sigma-field. If \(\sigma\le\tau\), then \(\mathcal F_\sigma\subseteq\mathcal F_\tau\). In discrete time, an adapted process \(X_n\) satisfies \(X_\tau1_{\{\tau<\infty\}}\in\mathcal F_\tau\).
Proof. Closure under complements follows from \[ A^c\cap\{\tau\le t\} =\{\tau\le t\}\setminus(A\cap\{\tau\le t\}), \] and countable unions commute with intersection by \(\{\tau\le t\}\). If \(A\in\mathcal F_\sigma\) and \(\sigma\le\tau\), then \[ A\cap\{\tau\le t\} =(A\cap\{\sigma\le t\})\cap\{\tau\le t\}\in\mathcal F_t. \] Finally, for a Borel set \(B\), \[ \{X_\tau\in B,\tau<\infty\}\cap\{\tau\le n\} =\bigcup_{k=0}^n\{X_k\in B,\tau=k\}\in\mathcal F_n. \]
Proposition (stopped process). If \(X\) is adapted and \(\tau\) is a stopping time, then \[ X_n^\tau:=X_{n\wedge\tau} \] is adapted in discrete time. The same holds in continuous time for an adapted process with right-continuous paths under the usual conditions.
This measurable framework is what makes expressions such as \(\mathbb E[M_\tau\mid\mathcal F_\sigma]\) meaningful. The next pages use it to formulate martingale transforms, maximal inequalities, and optional sampling without hidden anticipative assumptions.