First Hitting Times Of Continuous Adapted Processes

First Entry into a Closed Set

A stopping time records an event that can be determined using information available by the time in question. For a continuous adapted process, the first time its path reaches a closed set is a basic example. The key is to replace the exact condition “the process is in the set” with a distance that can be tested at countably many rational times. Continuity then recovers the exact hit, while right continuity of the filtration handles the distinction between hitting strictly before a time and hitting by that time.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\), where the filtration is increasing and right-continuous: \(\mathcal{F}_t=\bigcap_{s>t}\mathcal{F}_s\) for every \(t\ge0\). Let \(X=(X_t)_{t\ge0}\) be a real-valued adapted process whose paths are continuous for every \(\omega\in\Omega\). Adapted means that \(X_t\) is \(\mathcal{F}_t\)-measurable for each fixed \(t\). A random time \(\tau:\Omega\to[0,\infty]\) is a stopping time if \(\{\tau\le t\}\in\mathcal{F}_t\) for every \(t\ge0\).

Distance as a Measurable Test

For a nonempty set \(A\subseteq\mathbb{R}\), define its distance function by \(d(x,A)=\inf_{a\in A}|x-a|\). Closedness is important because it ensures that distance zero identifies membership, not just proximity.

Proof. For any \(x,y\in\mathbb{R}\) and \(a\in A\), the triangle inequality gives \(|x-a|\le |x-y|+|y-a|\). Taking the infimum over \(a\) yields \(d(x,A)\le |x-y|+d(y,A)\). Interchanging \(x\) and \(y\) gives \(|d(x,A)-d(y,A)|\le |x-y|\), so the distance function is continuous. If \(x\in A\), then \(d(x,A)=0\). Conversely, if \(d(x,A)=0\), the definition of infimum gives a sequence \(a_n\in A\) with \(|x-a_n|<1/n\). Thus \(a_n\to x\), and closedness of \(A\) implies \(x\in A\). \(\square\)

For a fixed time \(r\), distances at rational times up to \(r\) determine whether the process has hit \(A\) by \(r\). The endpoint \(r\) need not be rational: rational times approaching it from below suffice because the paths are continuous.

Proof. Fix an outcome \(\omega\). If the path hits \(A\) at some \(s\le r\), then \(d(X_s(\omega),A)=0\). If \(s<r\), choose rational times in \([0,r]\) converging to \(s\); if \(s=r\), choose rational times converging to \(r\) from below when \(r>0\). Continuity of the path and of the distance function shows that the corresponding distances tend to zero. For \(r=0\), the rational-time set includes \(0\), so the same conclusion holds directly. Hence \(D_r=0\).

In the other direction, if \(D_r=0\), choose rational \(q_n\in[0,r]\) such that \(d(X_{q_n}(\omega),A)<1/n\). Compactness of \([0,r]\) gives a subsequence converging to some \(s\in[0,r]\). By path continuity and continuity of distance, \(d(X_s(\omega),A)=0\). The first lemma implies \(X_s(\omega)\in A\), so \(\tau_A(\omega)\le r\). Finally, each \(d(X_q,A)\) is \(\mathcal{F}_q\)-measurable, and \(\mathcal{F}_q\subseteq\mathcal{F}_r\) for \(q\le r\). The infimum is over a countable set, so \(D_r\) is \(\mathcal{F}_r\)-measurable, including when \(r=0\). Therefore \(\{D_r=0\}\in\mathcal{F}_r\). \(\square\)

The Hitting-Time Theorem

Proof. The empty-set case gives \(\tau_A=\infty\), so \(\{\tau_A\le t\}=\varnothing\in\mathcal{F}_t\). Now suppose \(A\) is nonempty. For \(u>0\), the strict event can be written as \(\{\tau_A<u\}=\bigcup_{r\in\mathbb{Q}\cap[0,u)}\{\tau_A\le r\}\). Indeed, if \(\tau_A<u\), there is a rational \(r\) with \(\tau_A\le r<u\); the reverse implication follows from \(r<u\). By the preceding lemma, \(\{\tau_A\le r\}\in\mathcal{F}_r\subseteq\mathcal{F}_u\), so \(\{\tau_A<u\}\in\mathcal{F}_u\). For \(u=0\), this strict event is empty and is also measurable.

For each \(t\ge0\), the extended-real inequality \(\tau_A\le t\) holds exactly when \(\tau_A<t+1/n\) for every positive integer \(n\). Thus \(\{\tau_A\le t\}=\bigcap_{n=1}^{\infty}\{\tau_A<t+1/n\}\). Each event on the right belongs to \(\mathcal{F}_{t+1/n}\). Since the filtration is increasing, membership in all these sigma-fields implies membership in \(\bigcap_{n=1}^{\infty}\mathcal{F}_{t+1/n}=\mathcal{F}_t\); the last equality follows from right continuity, because the sequence \(t+1/n\) decreases to \(t\). Therefore \(\{\tau_A\le t\}\in\mathcal{F}_t\) for every \(t\), which is the stopping-time property. \(\square\)

The two stages in this argument have different roles. Rational times and continuity identify the hit using information available up to each fixed time. Right continuity then changes the strict-time events into the defining events \(\{\tau_A\le t\}\). In fact, for paths continuous at every outcome, the distance argument in the preceding lemma directly gives \(\{\tau_A\le t\}\in\mathcal{F}_t\), so right continuity is stronger than needed for this particular conclusion. The proof above makes explicit the common strict-time route and where the stated filtration condition enters.

Worked Applications

Distance-Neighborhood Approximation

The same distance function gives a useful pathwise approximation to the hitting time. For \(n\ge1\), define \(\tau_n=\inf\{t\ge0:d(X_t,A)<1/n\}\), using \(\inf\varnothing=\infty\). These are first entries into open neighborhoods of \(A\), rather than hits of \(A\) itself.

Proof. Fix \(n\). For \(u>0\), continuity implies \(\{\tau_n<u\}=\bigcup_{q\in\mathbb{Q}\cap[0,u)}\{d(X_q,A)<1/n\}\). If entry occurs before \(u\), continuity makes the strict inequality hold at nearby rational times; and if it holds at a rational time \(q<u\), then \(\tau_n\le q<u\). Each event in the union belongs to \(\mathcal{F}_q\subseteq\mathcal{F}_u\). Thus \(\{\tau_n<u\}\in\mathcal{F}_u\). Right continuity, using \(\{\tau_n\le t\}=\bigcap_{k\ge1}\{\tau_n<t+1/k\}\), gives \(\{\tau_n\le t\}\in\mathcal{F}_t\), including \(t=0\). Since \(1/(n+1)<1/n\), the \((n+1)\)st neighborhood is contained in the \(n\)th, and consequently \(\tau_n\le\tau_{n+1}\).

Now fix a continuous path. If \(\tau_A<\infty\), then \(d(X_{\tau_A},A)=0\), so every neighborhood is entered no later than \(\tau_A\); hence \(L=\lim_n\tau_n\le\tau_A\). If \(L<\infty\), then every \(\tau_n\) is finite. By the definition of an infimum, choose \(s_n\) with \(d(X_{s_n},A)<1/n\) and \(\tau_n\le s_n<\tau_n+1/n\). Since \(\tau_n\to L\), these inequalities give \(s_n\to L\). Continuity of the path and of the distance function now gives \(d(X_L,A)=0\). Closedness implies \(X_L\in A\), so \(\tau_A\le L\). If \(L=\infty\), then \(\tau_A\) cannot be finite, since a finite \(\tau_A\) would imply \(\tau_n\le\tau_A\) for all \(n\). Thus \(L=\tau_A\) in every case. \(\square\)

Why the Conditions Matter

Closedness prevents a sequence of process values from approaching a set without its limiting value belonging to that set. For example, \(A=(0,1)\) is not closed and \(x=0\) has distance zero from \(A\) but does not belong to \(A\). Continuity is what lets rational-time observations recover values at all times and lets a sequence of near-hits produce an actual hit on a compact time interval. Without it, rational observations alone need not determine whether the process visited the set at an irrational time.

In the proof, right continuity supplies the passage from \(\{\tau_A<u\}\) to \(\{\tau_A\le t\}\). The stronger direct distance characterization also shows that, when paths are continuous at every outcome and the process is adapted, those defining events are already measurable without that passage. If continuity is assumed only almost surely rather than for every outcome, one must take care with exceptional paths: changing a random time on a null set preserves measurability only when the relevant null sets belong to the filtration. This is one reason to state the path and filtration assumptions precisely.