2.11: Filtrations and Stopping Times
Introduction
Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process with state space \( (S, \mathscr{S}) \) defined on an underlying probability space \( (\Omega, \mathscr{F}, \P) \). To review, \( \Omega \) is the set of outcomes, \( \mathscr{F} \) the \( \sigma \)-algebra of events, and \( \P \) the probability measure on \( (S, \mathscr{S}) \). Also \( S \) is the set of states, and \( \mathscr{S} \) the \( \sigma \)-algebra of admissible subsets of \( S \). Usually, \( S \) is a topological space and \( \mathscr{S} \) the Borel \( \sigma \)-algebra generated by the open subsets of \( S \). A standard set of assumptions is that the topology is locally compact, Hausdorff, and has a countable base, which we will abbreviate by LCCB . For the index set, we assume that either \( T = \N \) or that \( T = [0, \infty) \) and as usual in these cases, we interpret the elements of \( T \) as points of time. The set \( T \) is also given a topology, the discrete topology in the first case and the standard Euclidean topology in the second case, and then the Borel \( \sigma \)-algebra \( \mathscr{T} \). So in discrete time with \( T = \N \), \( \mathscr{T} = \mathscr{P}(T) \), the power set of \( T \), so every subset of \( T \) is measurable, as is every function from \( T \) into a another measurable space. Finally, \( X_t \) is a random variable and so by definition is measurable with respect to \( \mathscr{F} \) and \( \mathscr{S} \) for each \( t \in T \). We interpret \( X_t \) is the state of some random system at time \( t \in T \). Many important concepts involving \( \bs{X} \) are based on how the future behavior of the process depends on the past behavior, relative to a given current time.
For \( t \in T \), let \( \mathscr{F}_t = \sigma\left\{X_s: s \in T, \; s \le t\right\} \), the \( \sigma \)-algebra of events that can be defined in terms of the process up to time \( t \). Roughly speaking, for a given \( A \in \mathscr{F}_t \), we can tell whether or not \( A \) has occurred if we are allowed to observe the process up to time \( t \). The family of \( \sigma \)-algebras \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) has two critical properties: the family is increasing in \( t \in T\), relative to the subset partial order, and all of the \( \sigma \)-algebras are sub \( \sigma \)-algebras of \( \mathscr{F} \). That is for \( s, \, t \in T \) with \( s \le t \), we have \( \mathscr{F}_s \subseteq \mathscr{F}_t \subseteq \mathscr{F} \).
Filtrations
Basic Definitions
Sometimes we need \( \sigma \)-algebras that are a bit larger than the ones in the last paragraph. For example, there may be other random variables that we get to observe, as time goes by, besides the variables in \( \bs{X} \). Sometimes, particularly in continuous time, there are technical reasons for somewhat different \( \sigma \)-algebras. Finally, we may want to describe how our information grows, as a family of \( \sigma \)-algebras, without reference to a random process. For the remainder of this section, we have a fixed measurable space \( (\Omega, \mathscr{F}) \) which we again think of as a sample space, and the time space \( (T, \mathscr{T}) \) as described above.
A family of \( \sigma \)-algebras \(\mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) if \(s, \, t \in T\) and \(s \le t\) imply \(\mathscr{F}_s \subseteq \mathscr{F}_t \subseteq \mathscr{F}\). The object \( \left(\Omega, \mathscr{F}, \mathfrak{F}\right) \) is a filtered sample space . If \( \P \) is a probability measure on \( (\Omega, \mathscr{F}) \), then \( \left(\Omega, \mathscr{F}, \mathfrak{F}, \P\right) \) is a filtered probability space .
So a filtration is simply an increasing family of sub-\(\sigma\)-algebras of \( \mathscr{F} \), indexed by \( T \). We think of \( \mathscr{F}_t \) as the \( \sigma \)-algebra of events up to time \( t \in T \). The larger the \( \sigma \)-algebras in a filtration, the more events that are available, so the following relation on filtrations is natural.
Suppose that \( \mathfrak{F} =\{\mathscr{F}_t: t \in T\} \) and \( \mathfrak{G} = \{\mathscr{G}_t: t \in T\} \) are filtrations on \( (\Omega, \mathscr{F}) \). We say that \( \mathfrak{F} \) is coarser than \( \mathfrak{G} \) and \( \mathfrak{G} \) is finer than \( \mathfrak{F} \), and we write \( \mathfrak{F} \preceq \mathfrak{G} \), if \( \mathscr{F}_t \subseteq \mathscr{G}_t \) for all \( t \in T \). The relation \( \preceq \) is a partial order on the collection of filtrations on \( (\Omega, \mathscr{F}) \). That is, if \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \), \( \mathfrak{G} = \{\mathscr{G}_t: t \in T\} \), and \( \mathfrak{H} = \mathscr{H}_t: t \in T\} \) are filtrations then
- \( \mathfrak{F} \preceq \mathfrak{F} \), the reflexive property .
- If \( \mathfrak{F} \preceq \mathfrak{G} \) and \( \mathfrak{G} \preceq \mathfrak{F} \) then \( \mathfrak{F} = \mathfrak{G} \), the antisymmetric property .
- If \( \mathfrak{F} \preceq \mathfrak{G} \) and \( \mathfrak{G} \preceq \mathfrak{H} \) then \( \mathfrak{F} \preceq \mathfrak{H} \), the transitive property .
Proof
The proof is a simple consequence of the fact that the subset relation defines a partial order.
- \( \mathscr{F}_t \subseteq \mathscr{F}_t \) for each \( t \in T \) so \( \mathfrak{F} \preceq \mathfrak{F} \).
- If \( \mathfrak{F} \preceq \mathfrak{G} \) and \( \mathfrak{G} \preceq \mathfrak{F} \) then \( \mathscr{F}_t \subseteq \mathscr{G}_t \) and \( \mathscr{G}_t \subseteq \mathscr{F}_t \) for each \( t \in T \). Hence \( \mathscr{F}_t = \mathscr{G}_t \) for each \( t \in T \) and so \( \mathfrak{F} = \mathfrak{G} \).
- If \( \mathfrak{F} \preceq \mathfrak{G} \) and \( \mathfrak{G} \preceq \mathfrak{H} \) then \( \mathscr{F}_t \subseteq \mathscr{G}_t \) and \( \mathscr{G}_t \subseteq \mathscr{H}_t \) for each \( t \in T \). Hence \( \mathscr{F}_t \subseteq \mathscr{H}_t \) for each \( t \in T \) and so \( \mathfrak{F} \preceq \mathfrak{H} \)
So the coarsest filtration on \( (\Omega, \mathscr{F}) \) is the one where \( \mathscr{F}_t = \{\Omega, \emptyset\} \) for every \( t \in T \) while the finest filtration is the one where \( \mathscr{F}_t = \mathscr{F} \) for every \( t \in T \). In the first case, we gain no information as time evolves, and in the second case, we have complete information from the beginning of time. Usually neither of these is realistic.
It's also natural to consider the \( \sigma \)-algebra that encodes our information over all time.
For a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) on \( (\Omega, \mathscr{F}) \), define \(\mathscr{F}_\infty = \sigma \left( \bigcup\left\{\mathscr{F}_t: t \in T\right\} \right)\). Then
- \(\mathscr{F}_\infty = \sigma \left( \bigcup\left\{\mathscr{F}_t: t \in T, t \ge s\right\} \right)\) for \( s \in T \).
- \( \mathscr{F}_t \subseteq \mathscr{F}_\infty \) for \( t \in T \).
Proof
These results follows since the \(\sigma\)-algebras in a filtration are increasing in time.
Of course, it may be the case that \( \mathscr{F}_\infty = \mathscr{F} \), but not necessarily. Recall that the intersection of a collection of \( \sigma \)-algebras on \( (\Omega, \mathscr{F}) \) is another \( \sigma \)-algebra. We can use this to create new filtrations from a collection of given filtrations.
Suppose that \( \mathfrak{F}_i = \left\{\mathscr{F}^i_t: t \in T\right\} \) is a filtration on \( (\Omega, \mathscr{F}) \) for each \( i \) in a nonempty index set \( I \). Then \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) where \( \mathscr{F}_t = \bigcap_{i \in I} \mathscr{F}^i_t \) for \( t \in T \) is also a filtration on \( (\Omega, \mathscr{F}) \). This filtration is sometimes denoted \( \mathfrak{F} = \bigwedge_{i \in I} \mathfrak{F}_i \), and is the finest filtration that is coarser than \( \mathfrak{F}_i \) for every \( i \in I \).
Proof
Suppose \( s, \, t \in T \) with \( s \le t \). Then \(\mathscr{F}^i_s \subseteq \mathscr{F}^i_t \subseteq \mathscr{F}\) for each \( i \in I \) so it follows that \( \bigcap_{i \in I} \mathscr{F}^i_s \subseteq \bigcap_{i \in I} \mathscr{F}^i_t \subseteq \mathscr{F} \).
Unions of \( \sigma \)-algebras are not in general \( \sigma \)-algebras, but we can construct a new filtration from a given collection of filtrations using unions in a natural way.
Suppose again that \( \mathfrak{F}_i = \left\{\mathscr{F}^i_t: t \in T\right\} \) is a filtration on \( (\Omega, \mathscr{F}) \) for each \( i \) in a nonempty index set \( I \). Then \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) where \( \mathscr{F}_t = \sigma\left(\bigcup_{i \in I} \mathscr{F}^i_t\right) \) for \( t \in T \) is also a filtration on \( (\Omega, \mathscr{F}) \). This filtration is sometimes denoted \( \mathfrak{F} = \bigvee_{i \in I} \mathfrak{F}_i \), and is the coarsest filtration that is finer than \( \mathfrak{F}_i \) for every \( i \in I \).
Proof
Suppose \( s, \, t \in T \) with \( s \le t \). Then \(\mathscr{F}^i_s \subseteq \mathscr{F}^i_t \subseteq \mathscr{F}\) for each \( i \in I \) so it follows that \( \bigcup_{i \in I} \mathscr{F}^i_s \subseteq \bigcup_{i \in I} \mathscr{F}^i_t \subseteq \mathscr{F} \), and hence \( \sigma\left(\bigcup_{i \in I} \mathscr{F}^i_s\right) \subseteq \sigma\left(\bigcup_{i \in I} \mathscr{F}^i_t\right) \subseteq \mathscr{F} \).
Stochastic Processes
Note again that we can have a filtration without an underlying stochastic process in the background. However, we usually do have a stochastic process \( \bs{X} = \{X_t: t \in T\} \), and in this case the filtration \( \mathfrak{F}^0 = \{\mathscr{F}^0_t: t \in T\} \) where \( \mathscr{F}^0_t = \sigma\{X_s: s \in T, \, s \le t\} \) is the natural filtration associated with \( \bs{X} \). More generally, the following definition is appropriate.
A stochastic process \(\bs{X} = \{X_t: t \in T\}\) on \( (\Omega, \mathscr{F}) \) is adapted to a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\}\) on \( (\Omega, \mathscr{F}) \) if \(X_t\) is measureable with respect to \(\mathscr{F}_t\) for each \(t \in T\).
Equivalently, \( \bs{X} \) is adapted to \( \mathfrak{F} \) if \( \mathfrak{F} \) is finer than \( \mathfrak{F}^0 \), the natural filtration associated with \( \bs{X} \). That is, \( \sigma\{X_s: s \in T, \; s \le t\} \subseteq \mathscr{F}_t \) for each \( t \in T \). So clearly, if \( \bs{X} \) is adapted to a filtration, then it is adapted to any finer filtration, and \( \mathfrak{F}^0 \) is the coarsest filtration to which \( \bs{X} \) is adapted. The basic idea behind the definition is that if the filtration \( \mathfrak{F} \) encodes our information as time goes by, then the process \( \bs{X} \) is observable. In discrete time, there is a related definition.
Suppose that \( T = \N \). A stochastic process \( \bs{X} = \{X_n: n \in \N\} \) is predictable by the filtration \( \mathfrak{F} = \{\mathscr{F}_n: n \in \N\} \) if \( X_{n +1}\) is measurable with respect to \( \mathscr{F}_n \) for all \( n \in \N \).
Clearly if \( \bs{X} \) is predictable by \( \mathfrak{F} \) then \( \bs{X} \) is adapted to \( \mathfrak{F} \). But predictable is better than adapted, in the sense that if \( \mathfrak{F} \) encodes our information as time goes by, then we can look one step into the future in terms of \( \bs{X} \): at time \( n \) we can determine \( X_{n+1} \). The concept of predictability can be extended to continuous time, but the definition is much more complicated.
Note that ultimately, a stochastic process \( \bs{X} = \{X_t: t \in T\} \) with sample space \( (\Omega, \mathscr{F}) \) and state space \( (S, \mathscr{S}) \) can be viewed a function from \( \Omega \times T \) into \( S \), so \( X_t(\omega) \in S \) is the state at time \( t \in T \) corresponding to the outcome \( \omega \in \Omega \). By definition, \( \omega \mapsto X_t(\omega) \) is measurable for each \( t \in T \), but it is often necessary for the process to be jointly measurable in \( \omega \) and \( t \).
Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process with sample space \( (\Omega, \mathscr{F}) \) and state space \( (S, \mathscr{S}) \). Then \( \bs{X} \) is measurable if \( \bs{X}: \Omega \times T \to S \) is measurable with respect to \( \mathscr{F} \otimes \mathscr{T} \) and \( \mathscr{S} \).
When we have a filtration, as we usually do, there is a stronger condition that is natural. Let \( T_t = \{s \in T: s \le t\} \) for \( t \in T \), and let \( \mathscr{T}_t = \{A \cap T_t: A \in \mathscr{T}\} \) be the corresponding induced \( \sigma \)-algebra.
Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process with sample space \( (\Omega, \mathscr{F}) \) and state space \( (S, \mathscr{S}) \), and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration. Then \( \bs{X} \) is progressively measurable relative to \( \mathfrak{F} \) if \( \bs{X}: \Omega \times T_t \to S\) is measurable with respect to \( \mathscr{F}_t \otimes \mathscr{T}_t \) and \( \mathscr{S} \) for each \( t \in T \).
Clearly if \( \bs{X} \) is progressively measurable with respect to a filtration, then it is progressively measurable with respect to any finer filtration. Of course when \( T \) is discrete , then any process \( \bs{X} \) is measurable, and any process \( \bs{X} \) adapted to \( \mathfrak{F} \) progressively measurable, so these definitions are only of interest in the case of continuous time.
Suppose again that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process with sample space \( (\Omega, \mathscr{F}) \) and state space \( (S, \mathscr{S}) \), and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration. If \( \bs{X} \) is progressively measurable relative to \( \mathfrak{F} \) then
- \( \bs{X} \) is measurable.
- \( \bs{X} \) is adapted to \( \mathfrak{F} \).
Proof
Suppose that \( \bs{X} \) is progressively measurable relative to \( \mathfrak{F} \).
- If \( A \in \mathscr{S} \) then \[ \bs{X}^{-1}(A) = \{(\omega, t) \in \Omega \times T: X_t(\omega) \in A\} = \bigcup_{n=1}^\infty \{(\omega, t) \in \Omega \times T_n: X_t(\omega) \in A\} \] By assumption, the \( n \)th term in the union is in \( \mathscr{F} \otimes \mathscr{T}_n \subseteq \mathscr{F} \otimes \mathscr{T} \), so the union is in \( \mathscr{F} \otimes \mathscr{T} \).
- Suppose that \( t \in T \). Then \( \bs{X}: \Omega \times T_t \to S\) is measurable with respect to \( \mathscr{F}_t \otimes \mathscr{T}_t \) and \( \mathscr{S} \). But \( X_t: \Omega \to S \) is just the cross section of this function at \( t \) and hence is measurable with respect to \( \mathscr{F}_t \) and \( \mathscr{S} \).
When the state space is a topological space (which is usually the case), then as you might guess, there is a natural link between continuity of the sample paths and progressive measurability.
Suppose that \( S \) has an LCCB topology and that \( \mathscr{S} \) is the \( \sigma \)-algebra of Borel sets. Suppose also that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is right continuous. Then \( \bs{X} \) is progressively measurable relative to the natural filtration \( \mathfrak{F}^0 \).
So if \( \bs{X} \) is right continuous, then \( \bs{X} \) is progressively measurable with respect to any filtration to which \( \bs{X} \) is adapted. Recall that in the previous section, we studied different ways that two stochastic processes can be equivalent. The following example illustrates some of the subtleties of processes in continuous time.
Suppose that \( \Omega = T = [0, \infty) \), \( \mathscr{F} = \mathscr{T} \) is the \( \sigma \)-algebra of Borel measurable subsets of \( [0, \infty) \), and \( \P \) is any continuous probability measure on \( (\Omega, \mathscr{F}) \). Let \( S = \{0, 1\} \) and \( \mathscr{S} = \mathscr{P}(S) = \{\emptyset, \{0\}, \{1\}, \{0, 1\}\} \). For \( t \in T \) and \( \omega \in \Omega \), define \( X_t(\omega) = \bs{1}_t(\omega) \) and \( Y_t(\omega) = 0 \). Then
- \( \bs{X} = \{X_t: t \in T\} \) is a version of \( \bs{Y} = \{Y_t: t \in T\} \)
- \( \bs{X} \) is not adapted to the natural filtration of \( \bs{Y} \).
Proof
- This was shown in the previous section, but here it is again: For \( t \in T \), \( \P(X_t \ne Y_t) = \P(\{t\}) = 0 \).
- Trivially, \( \sigma(Y_t) = \{\emptyset, \Omega\} \) for every \( t \in T \), so \( \sigma\{Y_s: 0 \le s \le t\} = \{\emptyset, \Omega\} \). But \( \sigma(X_t) = \{\emptyset, \{t\}, \Omega \setminus \{t\}, \Omega\} \).
Completion
Suppose now that \( P \) is a probability measure on \( (\Omega, \mathscr{F}) \). Recall that \(\mathscr{F}\) is complete with respect to \( P \) if \(A \in \mathscr{F}\), \( B \subseteq A \), and \(P(A) = 0\) imply \(B \in \mathscr{F}\) (and hence \( P(B) = 0 \)). That is, if \( A \) is an event with probability 0 and \( B \subseteq A \), then \( B \) is also an event (and also has probability 0). For a filtration, the following definition is appropriate.
The filtration \(\{\mathscr{F}_t: t \in T\}\) is complete with respect to a probability measure \( P \) on \( (\Omega, \mathscr{F}) \) if
- \(\mathscr{F}\) is complete with respect to \( P \)
- If \(A \in \mathscr{F}\) and \(P(A) = 0\) then \(A \in \mathscr{F}_0\).
Suppose \( P \) is a probability measure on \( (\Omega, \mathscr{F}) \) and that the filtration \(\{\mathscr{F}_t: t \in T\}\) is complete with respect to \( P \). If \(A \in \mathscr{F}\) is a null event (\(P(A) = 0\)) or an almost certain event (\(\P(A) = 1\)) then \(A \in \mathscr{F}_t\) for every \(t \in T\).
Proof
This follows since almost certain events are complements of null events and since the \(\sigma\)-algebras are increasing in \(t \in T\).
Recall that if \( P \) is a probability measure on \( (\Omega, \mathscr{F}) \), but \( \mathscr{F} \) is not complete with respect to \( P \), then \( \mathscr{F} \) can always be completed . Here's a review of how this is done: Let \[ \mathscr{N} = \{A \subseteq \Omega: \text{ there exists } N \in \mathscr{F} \text{ with } P(N) = 0 \text{ and } A \subseteq N\}\] So \( \mathscr{N} \) is the collection of null sets. Then we let \( \mathscr{F}^P = \sigma(\mathscr{F} \cup \mathscr{N}) \) and extend \( P \) to \( \mathscr{F}^P \) is the natural way: if \( A \in \mathscr{F}^P \) and \( A \) differs from \( B \in \mathscr{F} \) by a null set, then \( P(A) = P(B) \). Filtrations can also be completed.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \( P \) is a probability measure on \( (\Omega, \mathscr{F}) \). As above, let \( \mathscr{N} \) denote the collection of null subsets of \( \Omega \), and for \( t \in T \), let \( \mathscr{F}^P_t = \sigma(\mathscr{F}_t \cup \mathscr{N}) \).
Proof
If \( s, \, t \in T \) with \( s \le t \) then \( \mathscr{F}_s \subseteq \mathscr{F}_t \subseteq \mathscr{F} \) and hence \[ \sigma(\mathscr{F}_s \cup \mathscr{N}) \subseteq \sigma(\mathscr{F}_t \cup \mathscr{N}) \subseteq \sigma(\mathscr{F} \cup \mathscr{N})\] and so \( \mathscr{F}^P_s \subseteq \mathscr{F}^P_t \subseteq \mathscr{F}^P \). The probability measure \( P \) can be extended to \( \mathscr{F}^P \) as described above, and hence is defined on \( \mathscr{F}^P_t \) for each \( t \in T \). By construction, if \( A \in \mathscr{F}^P \) and \( P(A) = 0 \) then \( A \in \mathscr{F}^P_0 \) so \( \mathfrak{F}^P \) is complete with respect to \( P \).
Naturally, \( \mathfrak{F}^P \) is the completion of \( \mathfrak{F} \) with respect to \( P \). Sometimes we need to consider all probability measures on \( (\Omega, \mathscr{F}) \).
Let \( \mathscr{P} \) denote the collection of probability measures on \( (\Omega, \mathscr{F}) \), and suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \). Let \( \mathscr{F}^* = \bigcap \{\mathscr{F}^P: P \in \mathscr{P}\} \), and let \( \mathfrak{F}^* = \bigwedge \{\mathfrak{F}^P: P \in \mathscr{P}\} \). Then \( \mathfrak{F}^* \) is a filtration on \( (\Omega, \mathscr{F}^*) \), known as the universal completion of \( \mathfrak{F} \).
Proof
Note that \( \mathfrak{F}^P \) is a filtration on \( (\Omega, \mathscr{F}^P) \) for each \( P \in \mathscr{P} \), so \( \mathfrak{F}^* \) is a filtration on \( (\Omega, \mathscr{F}^*) \).
The last definition must seem awfully obscure, but it does have a place. In the theory of Markov processes, we usually allow arbitrary initial distributions, which in turn produces a large collection of distributions on the sample space.
Right Continuity
In continuous time, we sometimes need to refine a given filtration somewhat.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is a filtration on \( (\Omega, \mathscr{F}) \). For \( t \in [0, \infty) \), define \( \mathscr{F}_{t+} = \bigcap \{\mathscr{F}_s: s \in (t, \infty)\} \). Then \( \mathfrak{F}_+ = \{\mathscr{F}_{t+}: t \in T\} \) is also a filtration on \( (\Omega, \mathscr{F}) \) and is finer than \( \mathfrak{F} \).
Proof
For \( t \in [0, \infty) \) note that \( \mathscr{F}_{t+} \) is a \( \sigma \)-algebra since it is the intersection of \( \sigma \)-algebras, and clearly \( \mathscr{F}_{t+} \subseteq \mathscr{F} \). Next, if \( s, \, t \in [0, \infty) \) with \( s \le t \), then \( \{\mathscr{F}_r: r \in (t, \infty)\} \subseteq \{\mathscr{F}_r: r \in (s, \infty)\} \), so it follows that \[ \mathscr{F}_{s+} = \bigcap\{\mathscr{F}_r: r \in (s, \infty)\} \subseteq \bigcap\{\mathscr{F}_r: r \in (t, \infty)\} = \mathscr{F}_{t+} \] Finally, for \( t \in [0, \infty) \), \( \mathscr{F}_t \subseteq \mathscr{F}_s \) for every \( s \in (t, \infty) \) so \( \mathscr{F}_t \subseteq \bigcap\{\mathscr{F}_s: s \in (t, \infty)\} = \mathscr{F}_{t+} \).
Since the \( \sigma \)-algebras in a filtration are increasing, it follows that for \( t \in [0, \infty) \), \( \mathscr{F}_{t+} = \bigcap\{\mathscr{F}_s: s \in (t, t + \epsilon)\} \) for every \( \epsilon \in (0, \infty) \). So if the filtration \( \mathfrak{F} \) encodes the information available as time goes by, then the filtration \( \mathfrak{F}_+ \) allows an
infinitesimal peak into the future
at each \( t \in [0, \infty) \). In light of the previous result, the next definition is natural.
A filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is right continuous if \( \mathfrak{F}_+ = \mathfrak{F} \), so that \( \mathscr{F}_{t+} = \mathscr{F}_t \) for every \( t \in [0, \infty) \).
Right continuous filtrations have some nice properties, as we will see later. If the original filtration is not right continuous, the slightly refined filtration is:
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is a filtration. Then \( \mathfrak{F}_+ \) is a right continuous filtration.
Proof
For \( t \in T \) \[ \mathscr{F}_{t++} = \bigcap\{\mathscr{F}_{s+}: s \in (t, \infty)\} = \bigcap\left\{\bigcap\{\mathscr{F}_r: r \in (s, \infty)\}: s \in (t, \infty)\right\} = \bigcap\{\mathscr{F}_u: u \in (t, \infty)\} = \mathscr{F}_{t+} \]
For a stochastic process \( \bs{X} = \{X_t: t \in [0, \infty)\} \) in continuous time, often the filtration \( \mathfrak{F} \) that is most useful is the right-continuous refinement of the natural filtration. That is, \( \mathfrak{F} = \mathfrak{F}^0_+ \), so that \( \mathscr{F}_t = \sigma\{X_s: s \in [0, t]\}_+ \) for \( t \in [0, \infty) \).
Stopping Times
Basic Properties
Suppose again that we have a fixed sample space \( (\Omega, \mathscr{F}) \). Random variables taking values in the time set \( T \) are important, but often as we will see, it's necessary to allow such variables to take the value \( \infty \) as well as finite times. So let \( T_\infty = T \cup \{\infty\} \). We extend order to \( T_\infty \) by the obvious rule that \( t \lt \infty \) for every \( t \in T \). We also extend the topology on \( T \) to \( T_\infty \) by the rule that for each \( s \in T \), the set \( \{t \in T_\infty: t \gt s\} \) is an open neighborhood of \( \infty\). That is, \( T_\infty \) is the one-point compactification of \( T \). The reason for this is to preserve the meaning of time converging to infinity . That is, if \( (t_1, t_2, \ldots) \) is a sequence in \( T_\infty \) then \( t_n \to \infty \) as \( n \to \infty \) if and only if, for every \( t \in T \) there exists \( m \in \N_+ \) such that \( t_n \gt t \) for \( n \gt m \). We then give \( T_\infty \) the Borel \( \sigma \)-algebra \( \mathscr{T}_\infty \) as before. In discrete time, this is once again the discrete \( \sigma \)-algebra, so that all subsets are measurable. In both cases, we now have an enhanced time space is \( (T_\infty, \mathscr{T}_\infty) \). A random variable \( \tau \) taking values in \( T_\infty \) is called a random time .
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \). A random time \( \tau \) is a stopping time relative to \( \mathfrak{F} \) if \( \{\tau \le t\} \in \mathscr{F}_t \) for each \( t \in T \).
In a sense, a stopping time is a random time that does not require that we see into the future. That is, we can tell whether or not \( \tau \le t \) from our information at time \( t \). The term stopping time comes from gambling. Consider a gambler betting on games of chance. The gambler's decision to stop gambling at some point in time and accept his fortune must define a stopping time. That is, the gambler can base his decision to stop gambling on all of the information that he has at that point in time, but not on what will happen in the future. The terms Markov time and optional time are sometimes used instead of stopping time. If \( \tau \) is a stopping time relative to a filtration, then it is also a stoping time relative to any finer filtration:
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) and \( \mathfrak{G} = \{\mathscr{G}_t: t \in T\} \) are filtrations on \( (\Omega, \mathscr{F}) \), and that \(\mathfrak{G}\) is finer than \( \mathfrak{F} \). If a random time \( \tau \) is a stopping time relative to \( \mathfrak{F} \) then \( \tau \) is a stopping time relative to \( \mathfrak{G} \).
Proof
This is very simple. If \( t \in T \) then \( \{\tau \le t\} \in \mathscr{F}_t \) and hence \( \{\tau \le t\} \in \mathscr{G}_t \) since \( \mathscr{F}_t \subseteq \mathscr{G}_t \).
So, the finer the filtration, the larger the collection of stopping times. In fact, every random time is a stopping time relative to the finest filtration \( \mathfrak{F} \) where \( \mathscr{F}_t = \mathscr{F} \) for every \( t \in T \). But this filtration corresponds to having complete information from the beginning of time, which of course is usually not sensible. At the other extreme, for the coarsest filtration \( \mathfrak{F} \) where \( \mathscr{F}_t = \{\Omega, \emptyset\} \) for every \( t \in T \), the only stopping times are constants. That is, random times of the form \( \tau(\omega) = t \) for every \( \omega \in \Omega \), for some \(t \in T_\infty \).
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \). A random time \( \tau \) is a stopping time relative to \( \mathfrak{F} \) if and only if \( \{\tau \gt t\} \in \mathscr{F}_t \) for each \( t \in T \).
Proof
This result is trivial since \( \{\tau \gt t\} = \{\tau \le t\}^c \) for \( t \in T \).
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \), and that \( \tau \) is a stopping time relative to \( \mathfrak{F} \). Then
- \( \{\tau \lt t\} \in \mathscr{F}_t \) for every \( t \in T \).
- \( \{\tau \ge t\} \in \mathscr{F}_t\) for every \( t \in T \).
- \( \{\tau = t\} \in \mathscr{F}_t \) for every \( t \in T \).
Proof
- Suppose first that \( T = \N \). Then \( \{\tau \lt t\} = \{\tau \le t - 1\} \in \mathscr{F}_{t-1} \subseteq \mathscr{F}_t \) for \( t \in \N \). Next suppose that \( T = [0, \infty) \). Fix \(t \in (0, \infty)\) and let \((s_1, s_2, \ldots)\) be a strictly increasing sequence in \([0, \infty)\) with \(s_n \uparrow t\) as \(n \to \infty\). Then \( \{\tau \lt t\} = \bigcup_{n=1}^\infty \{\tau \le s_n\} \). But \( \{\tau \le s_n\} \in \mathscr{F}_{s_n} \subseteq \mathscr{F}_t \) for each \( n \), so \( \{\tau \lt t\} \in \mathscr{F}_t \).
- This follows from (a) since \( \{\tau \ge t\} = \{\tau \lt t\}^c \) for \( t \in T \).
- For \( t \in T \) note that \( \{\tau = t\} = \{\tau \le t\} \setminus \{\tau \lt t\} \). Both events in the set difference are in \( \mathscr{F}_t \).
Note that when \(T = \N\), we actually showed that \(\{\tau \lt t\} \in \mathscr{F}_{t-1}\) and \(\{\tau \ge t\} \in \mathscr{F}_{t-1}\). The converse to part (a) (or equivalently (b)) is not true, but in continuous time there is a connection to the right-continuous refinement of the filtration.
Suppose that \( T = [0, \infty) \) and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is a filtration on \( (\Omega, \mathscr{F}) \). A random time \( \tau \) is a stopping time relative to \( \mathfrak{F}_+ \) if and only if \( \{\tau \lt t\} \in \mathscr{F}_t \) for every \(t \in [0, \infty)\).
Proof
So restated, we need to show that \( \{\tau \le t\} \in \mathscr{F}_{t+} \) for every \( t \in [0, \infty) \) if and only if \( \{\tau \lt t\} \in \mathscr{F}_t \) for every \( t \in [0, \infty) \). (Note by the way, that this not the same as the statement that for every \( t \in T \), \( \{\tau \lt t\} \in \mathscr{F}_{t+} \) if and only if \( \{\tau \le t\} \in \mathscr{F}_t \), which is not true.) Suppose first that \( \tau \) is a stopping time relative to \( \mathfrak{F} \). Fix \(t \in [0, \infty)\) and let \((t_1, t_2, \ldots)\) be a strictly decreasing sequence in \([0, \infty)\) with \(t_n \downarrow t\) as \(n \to \infty\). Then for each \(k \in \N_+\), \( \{\tau \le t\} = \bigcap_{n=k}^\infty \{\tau \lt t_n\} \). If \( s \gt t \) then there exists \( k \in \N_+ \) such that \( t_n \lt s \) for each \( n \ge k \). Hence \( \{\tau \lt t_n\} \in \mathscr{F}_{t_n} \subseteq \mathscr{F}_s \) for \( n \ge k \), and so it follows that \( \{\tau \le t\} \in \mathscr{F}_s \). Since this is true for every \( s \gt t \) it follows \(\{\tau \le t\} \in \mathscr{F}_{t+}\). Conversely, suppose that \( \{\tau \le t\} \in \mathscr{F}_{t+} \) for every \( t \in [0, \infty) \). Fix \( t \in (0, \infty) \) and let \( (t_1, t_2, \ldots) \) be a strictly increasing sequence in \( (0, \infty) \) with \( t_n \uparrow t \) as \( n \to \infty \). Then \( \bigcup_{i=1}^\infty \{\tau \le t_n\} = \{\tau \lt t\} \). But for every \( n \in \N_+ \) \[ \{\tau \le t_n\} \in \mathscr{F}_{t_n+} = \bigcap\left\{\mathscr{F}_s: s \in (t_n, t)\right\} \subseteq \mathscr{F}_t \] Hence \( \{\tau \lt t \} \in \mathscr{F}_t\).
If \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is a filtration and \( \tau \) is a random time that satisfies \( \{\tau \lt t \} \in \mathscr{F}_t \) for every \( t \in T \), then some authors call \( \tau \) a weak stopping time or say that \( \tau \) is weakly optional for the filtration \( \mathfrak{F} \). But to me, the increase in jargon is not worthwhile, and it's better to simply say that \( \tau \) is a stopping time for the filtration \(\mathfrak{F}_+\). The following corollary now follows.
Suppose that \( T = [0, \infty) \) and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \) is a right-continuous filtration. A random time \( \tau \) is a stopping time relative to \( \mathfrak{F} \) if and only if \( \{\tau \lt t\} \in \mathscr{F}_t \) for every \( t \in [0, \infty) \).
The converse to part (c) of the result above holds in discrete time.
Suppose that \( T = \N \) and that \( \mathfrak{F} = \{\mathscr{F}_n: n \in \N\} \) is a filtration on \( (\Omega, \mathscr{F}) \). A random time \( \tau \) is a stopping time for \( \mathfrak{F} \) if and only if \( \{\tau = n\} \in \mathscr{F}_n \) for every \( n \in \N \).
Proof
If \(\tau\) is a stopping time then as shown above , \(\{\tau = n\} \in \mathscr{F}_n\) for every \( n \in \N \). Conversely, suppose that this condition holds. For \(n \in \N\), \(\{\tau \le n\} = \bigcup_{k=0}^n \{\tau = k\}\). But \(\{\tau = k\} \in \mathscr{F}_k \subseteq \mathscr{F}_n\) for \(k \in \{0, 1, \ldots, n\}\) so \(\{\tau \le n\} \in \mathscr{F}_n\).
Basic Constructions
As noted above, a constant element of \(T_\infty\) is a stopping time, but not a very interesting one.
Suppose \(s \in T_\infty\) and that \(\tau(\omega) = s\) for all \(\omega \in \Omega\). The \(\tau\) is a stopping time relative to any filtration on \( (\Omega, \mathscr{F}) \).
Proof
For \( t \in T \) note that \(\{\tau \le t\} = \Omega\) if \(s \le t\) and \(\{\tau \le t\} = \emptyset\) if \(s \gt t\).
If the filtration \(\{\mathscr{F}_t: t \in T\}\) is complete, then a random time that is almost certainly a constant is also a stopping time. The following theorems give some basic ways of constructing new stopping times from ones we already have.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \(\tau_1\) and \(\tau_2\) are stopping times relative to \( \mathfrak{F} \). Then each of the following is also a stopping time relative to \( \mathfrak{F} \):
- \(\tau_1 \vee \tau_2 = \max\{\tau_1, \tau_2\}\)
- \(\tau_1 \wedge \tau_2 = \min\{\tau_1, \tau_2\}\)
- \(\tau_1 + \tau_2\)
Proof
- Note that \(\{\tau_1 \vee \tau_2 \le t\} = \{\tau_1 \le t\} \cap \{\tau_2 \le t\} \in \mathscr{F}_t\) for \(t \in T\), so the result follows from the definition.
- Note that \(\{\tau_1 \wedge \tau_2 \gt t\} = \{\tau_1 \gt t\} \cap \{\tau_2 \gt t\} \in \mathscr{F}_t\) for \(t \in T\), so the result follows from the result above .
- This is simple when \(T = \N\). In this case, \(\{\tau_1 + \tau_2 \le t\} = \bigcup_{n=0}^t \{\tau_1 = n\} \cap \{\tau_2 \le t - n\}\). But for \(n \le t\), \(\{\tau_1 = n\} \in \mathscr{F}_n \subseteq \mathscr{F}_t\) and \(\{\tau_2 \le t - n\} \in \mathscr{F}_{t - n} \subseteq \mathscr{F}_t\). Hence \(\{\tau_1 + \tau_2 \le t\} \in \mathscr{F}_t\). Suppose instead that \(T = [0, \infty)\) and \(t \in T\). Then \(\tau_1 + \tau_2 \gt t\) if and only if either \(\tau_1 \le t\) and \(\tau_2 \gt t - \tau_1\) or \(\tau_1 \gt t\). Of course \(\{\tau_1 \gt t\} \in \mathscr{F}_t\) so we just need to show that the first event is also in \(\mathscr{F}_t\). Note that \(\tau_1 \le t\) and \(\tau_2 \gt t - \tau_1\) if and only if there exists a rational \(q \in [0, t]\) such that \(q \le \tau_1 \le t\) and \(\tau_2 \ge t - q\). Each of these events is in \(\mathscr{F}_t\) and hence so is the union of the events over the countable collection of rational \(q \in [0, t]\).
It follows that if \( (\tau_1, \tau_2, \ldots, \tau_n) \) is a finite sequence of stopping times relative to \( \mathfrak{F} \), then each of the following is also a stopping time relative to \( \mathfrak{F} \):
- \( \tau_1 \vee \tau_2 \vee \cdots \vee \tau_n \)
- \( \tau_1 \wedge \tau_2 \wedge \cdots \wedge \tau_n \)
- \( \tau_1 + \tau_2 + \cdots + \tau_n \)
We have to be careful when we try to extend these results to infinite sequences.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \), and that \((\tau_n: n \in \N_+)\) is a sequence of stopping times relative to \( \mathfrak{F} \). Then \(\sup\{\tau_n: n \in \N_+\}\) is also a stopping time relative to \( \mathfrak{F} \).
Proof
Let \(\tau = \sup\{\tau_n: n \in \N_+\}\). Note that \(\tau\) exists in \(T_\infty\) and is a random time. For \(t \in T\), \(\{\tau \le t\} = \bigcap_{n=1}^\infty \{\tau_n \le t\}\). But each event in the intersection is in \(\mathscr{F}_t\) and hence so is the intersection.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \), and that \((\tau_n: n \in \N_+)\) is an increasing sequence of stopping times relative to \( \mathfrak{F} \). Then \(\lim_{n \to \infty} \tau_n\) is a stopping time relative to \( \mathfrak{F} \).
Proof
This is a corollary of the previous theorem. Since the sequence is increasing, \(\lim_{n \to \infty} \tau_n = \sup\{\tau_n: n \in \N_+\}\).
Suppose that \( T = [0, \infty) \) and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \). If \((\tau_n: n \in \N_+)\) is a sequence of stopping times relative to \( \mathfrak{F} \), then each of the following is a stopping time relative to \( \mathfrak{F}_+ \):
- \(\inf\left\{\tau_n: n \in \N_+\right\}\)
- \(\liminf_{n \to \infty} \tau_n\)
- \(\limsup_{n \to \infty} \tau_n\)
Proof
- Let \(\tau = \inf\left\{\tau_n: n \in \N_+\right\}\). Then \(\{\tau \ge t\} = \bigcap_{n=1}^\infty\{\tau_n \ge t\} \in \mathscr{F}_t\) for \(t \in T\). Hence \(\tau\) is a stopping time relative to \( \mathfrak{F}_+ \) by the result above.
- Recall that \(\liminf_{n \to \infty} \tau_n = \sup\left\{\inf\{\tau_k: k \ge n\}: n \in \N_+\right\}\) and so this is a stopping time relative to \( \mathfrak{F}_+ \) by part (a) and the result above on supremums.
- Similarly note that \(\limsup_{n \to \infty} \tau_n = \inf\left\{\sup\{\tau_k: k \ge n\}: n \in \N_+\right\}\) and so this is a stopping time relative to \( \mathfrak{F}_+ \) by part (a) and the result above on supremums.
As a simple corollary, we have the following results:
Suppose that \( T = [0, \infty) \) and that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a right-continuous filtration on \( (\Omega, \mathscr{F}) \). If \((\tau_n: n \in \N_+)\) is a sequence of stopping times relative to \( \mathfrak{F} \), then each of the following is a also a stopping time relative to \( \mathfrak{F} \):
- \(\inf\left\{\tau_n: n \in \N_+\right\}\)
- \(\liminf_{n \to \infty} \tau_n\)
- \(\limsup_{n \to \infty} \tau_n\)
The \(\sigma\)-Algebra of a Stopping Time
Consider again the general setting of a filtration \(\mathfrak{F} = \{\mathscr{F}_t: t \in T\}\) on the sample space \((\Omega, \mathscr{F})\), and suppose that \(\tau\) is a stopping time relative to \( \mathfrak{F} \). We want to define the \(\sigma\)-algebra \(\mathscr{F}_\tau\) of events up to the random time \(\tau\), analagous to \(\mathscr{F}_t\) the \( \sigma \)-algebra of events up to a fixed time \(t \in T\). Here is the appropriate definition:
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \( \tau \) is a stopping time relative to \( \mathfrak{F} \). Define \( \mathscr{F}_\tau = \left\{A \in \mathscr{F}: A \cap \{\tau \le t\} \in \mathscr{F}_t \text{ for all } t \in T\right\} \). Then \( \mathscr{F}_\tau \) is a \( \sigma \)-algebra.
Proof
First \(\Omega \in \mathscr{F}_\tau\) since \(\Omega \cap \{\tau \le t\} = \{\tau \le t\} \in \mathscr{F}_t\) for \(t \in T\). If \(A \in \mathscr{F}_\tau\) then \(A^c \cap \{\tau \le t\} = \{\tau \le t \} \setminus \left(A \cap \{\tau \le t\}\right) \in \mathscr{F}_t\) for \(t \in T\). Finally, suppose that \(A_i \in \mathscr{F}_\tau\) for \(i\) in a countable index set \(I\). Then \(\left(\bigcup_{i \in I} A_i\right) \cap \{\tau \le t\} = \bigcup_{i \in I} \left(A_i \cap \{\tau \le t\}\right) \in \mathscr{F}_t\) for \(t \in T\).
Thus, an event \(A\) is in \(\mathscr{F}_\tau\) if we can determine if \(A\) and \(\tau \le t\) both occurred given our information at time \(t\). If \(\tau\) is constant, then \(\mathscr{F}_\tau\) reduces to the corresponding member of the original filtration, which clealry should be the case, and is additional motivation for the definition.
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \). Fix \(s \in T\) and define \(\tau(\omega) = s\) for all \(\omega \in \Omega\). Then \(\mathscr{F}_\tau = \mathscr{F}_s\).
Proof
Suppose that \(A \in \mathscr{F}_s\). Then \(A \in \mathscr{F}\) and for \(t \in T\), \(A \cap \{\tau \le t\} = A\) if \(s \le t\) and \(A \cap \{\tau \le t\} = \emptyset\) if \(s \gt t\). In either case, \(A \cap \{\tau \le t\} \in \mathscr{F}_t\) and hence \(A \in \mathscr{F}_\tau\). Conversely, suppose that \(A \in \mathscr{F}_\tau\). Then \(A = A \cap \{\tau \le s\} \in \mathscr{F}_s\).
Clearly, if we have the information available in \(\mathscr{F}_\tau\), then we should know the value of \(\tau\) itself. This is also true:
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \( \tau \) is a stopping time relative to \( \mathfrak{F} \). Then \(\tau\) is measureable with respect to \(\mathscr{F}_\tau\).
Proof
It suffices to show that \(\{\tau \le s\} \in \mathscr{F}_\tau\) for each \(s \in T\). For \( s, \, t \in T \), \[\{\tau \le t\} \cap \{\tau \le s\} = \{\tau \le s \wedge t\} \in \mathscr{F}_{s \wedge t} \subseteq \mathscr{F}_t\] Hence \(\{\tau \le s\} \in \mathscr{F}_\tau\).
Here are other results that relate the \( \sigma \)-algebra of a stopping time to the original filtration.
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \( \tau \) is a stopping time relative to \( \mathfrak{F} \). If \(A \in \mathscr{F}_\tau\) then for \(t \in T\),
- \(A \cap \{\tau \lt t\} \in \mathscr{F}_t\)
- \(A \cap \{\tau = t\} \in \mathscr{F}_t\)
Proof
- By definition, \(A \cap \{\tau \le t\} \in \mathscr{F}_t\). But \(\{\tau \lt t\} \subseteq \{\tau \le t\}\) and \(\{\tau \lt t\} \in \mathscr{F}_t\). Hence \(A \cap \{\tau \lt t\} = A \cap \{\tau \le t\} \cap \{\tau \lt t\} \in \mathscr{F}_t\).
- similarly \(\{\tau = t\} \subseteq \{\tau \le t\}\) and \(\{\tau = t\} \in \mathscr{F}_t\). Hence \(A \cap \{\tau = t\} = A \cap \{\tau \le t\} \cap \{\tau = t\} \in \mathscr{F}_t\)
The \( \sigma \)-algebra of a stopping time relative to a filtration is related to the \( \sigma \)-algebra of the stopping time relative to a finer filtration in the natural way.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) and \( \mathfrak{G} = \{\mathscr{G}_t: t \in T\} \) are filtrations on \( (\Omega, \mathscr{F}) \) and that \( \mathfrak{G} \) is finer than \( \mathfrak{F} \). If \( \tau \) is a stopping time relative to \( \mathfrak{F} \) then \( \mathscr{F}_\tau \subseteq \mathscr{G}_\tau \).
Proof
From the result above, \( \tau \) is also a stopping time relative to \( \mathfrak{G} \), so the statement makes sense. If \( A \in \mathscr{F}_\tau \) then for \( t \in T \), \( A \cap \{\tau \le t\} \in \mathscr{F}_t \subseteq \mathscr{G}_t \), so \( A \in \mathscr{G}_\tau \).
When two stopping times are ordered, their \( \sigma \)-algebras are also ordered.
Suppose that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \) and that \(\rho\) and \(\tau\) are stopping times for \( \mathfrak{F} \) with \(\rho \le\tau\). Then \(\mathscr{F}_\rho \subseteq \mathscr{F}_\tau\).
Proof
Suppose that \(A \in \mathscr{F}_\rho\) and \(t \in T\). Note that \(\{\tau \le t\} \subseteq \{\rho \le t\}\). By definition, \(A \cap \{\rho \le t\} \in \mathscr{F}_t\) and \(\{\tau \le t\} \in \mathscr{F}_t \). Hence \(A \cap \{\tau \le t\} = A \cap \{\rho \le t\} \cap \{\tau \le t\} \in \mathscr{F}_t\), so \(A \in \mathscr{F}_\tau\).
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \), and that \(\rho\) and \(\tau\) are stopping times for \( \mathfrak{F} \). Then each of the following events is in \(\mathscr{F}_\tau\) and in \(\mathscr{F}_\rho\).
- \(\{\rho \lt \tau\}\)
- \(\{\rho = \tau\}\)
- \(\{\rho \gt \tau\}\)
- \(\{\rho \le \tau\}\)
- \(\{\rho \ge \tau\}\)
Proof
The proofs are easy when \(T = \N\).
- Let \(t \in T\). Then \[\{\rho \lt \tau\} \cap \{\tau \le t\} = \bigcup_{n=0}^t \bigcup_{k=0}^{n-1} \{\tau = n, \rho = k\}\] But each event in the union is in \(\mathscr{F}_t\).
- Similarly, let \(t \in T\). Then \[ \{\rho = \tau\} \cap \{\tau \le t\} = \bigcup_{n=0}^t \{\rho = n, \tau = n\} \] and again each event in the union is in \(\mathscr{F}_t\).
- This follows from symmetry, reversing the roles of \(\rho\) and \(\tau\) in part (a).
- Note that \(\{\rho \le \tau\} = \{\rho \lt \tau\} \cup \{\rho = \tau\} \in \mathscr{F}_\tau\).
- Similarly, note that \(\{\rho \ge \tau\} = \{\rho \gt \tau\} \cup \{\rho = \tau\} \in \mathscr{F}_\tau\).
We can
stop
a filtration at a stopping time. In the next subsection, we will stop a stochastic process in the same way.
Suppose again that \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a filtration on \( (\Omega, \mathscr{F}) \), and that \(\tau\) is a stopping times for \( \mathfrak{F} \). For \( t \in T \) define \( \mathscr{F}^\tau_t = \mathscr{F}_{t \wedge \tau} \). Then \( \mathfrak{F}^\tau = \{\mathscr{F}^\tau_t: t \in T\} \) is a filtration and is coarser than \( \mathfrak{F} \).
Proof
The random time \( t \wedge \tau \) is a stopping time for each \( t \in T \) by the result above, so \( \mathscr{F}^\tau_t \) is a sub \( \sigma \)-algebra of \( \mathscr{F} \). If \( t \in T \), then by definition, \( A \in \mathscr{F}^\tau_t \) if and only if \( A \cap \{t \wedge \tau \le r\} \in \mathscr{F}_r \) for every \( r \in T \). But for \( r \in T \), \( \{t \wedge \tau \le r\} = \Omega \) if \( r \ge t \) and \( \{t \wedge \tau \le r\} = \{\tau \le r\} \) if \( r \lt t \). Hence \( A \in \mathscr{F}^\tau_t \) if and only if \( A \cap \{\tau \le r\} \in \mathscr{F}_r \) for \( r \lt t \) and \( A \in \mathscr{F}_t \). So in particular, \( \mathfrak{F}^\tau \) is coarser than \( \mathfrak{F} \). Further, suppose \( s, \, t \in T \) with \( s \le t \), and that \( A \in \mathscr{F}^\tau_s \). Let \( r \in T \). If \( r \lt s \) then \( A \cap \{\tau \le r\} \in \mathscr{F}_r \). If \( s \le r \lt t \) then \( A \in \mathscr{F}_s \subseteq \mathscr{F}_r \) and \( \{\tau \le r\} \in \mathscr{F}_r \) so again \( A \cap \{\tau \le r\} \in \mathscr{F}_r \). Finally if \( r \ge t \) then \( A \in \mathscr{F}_s \subseteq \mathscr{F}_t \). Hence \( A \in \mathscr{F}^\tau_t \).
Stochastic Processes
As usual, the most common setting is when we have a stochastic process \( \bs{X} = \{X_t: t \in T\} \) defined on our sample space \( (\Omega, \mathscr{F}) \) and with state space \( (S, \mathscr{S}) \). If \( \tau \) is a random time, we are often interested in the state \( X_\tau \) at the random time. But there are two issues. First, \( \tau \) may take the value infinity, in which case \( X_\tau \) is not defined. The usual solution is to introduce a new
death state
\( \delta \), and define \( X_\infty = \delta \). The \( \sigma \)-algebra \( \mathscr{S} \) on \( S \) is extended to \( S_\delta = S \cup \{\delta\} \) in the natural way, namely \( \mathscr{S}_\delta = \sigma(S \cup \{\delta\}) \).
Our other problem is that we naturally expect \( X_\tau \) to be a random variable (that is, measurable), just as \( X_t \) is a random variable for a deterministic \( t \in T \). Moreover, if \( \bs{X} \) is adapted to a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \), then we would naturally also expect \( X_\tau \) to be measurable with respect to \( \mathscr{F}_\tau \), just as \( X_t \) is measurable with respect to \( \mathscr{F}_t \) for deterministic \( t \in T \). But this is not obvious, and in fact is not true without additional assumptions. Note that \( X_\tau \) is a random state at a random time, and so depends on an outcome \( \omega \in \Omega \) in two ways: \(X_{\tau(\omega)}(\omega)\).
Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on the sample space \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \), and that \( \bs{X} \) is measurable. If \( \tau \) is a finite random time, then \( X_\tau \) is measurable. That is, \( X_\tau \) is a random variable with values in \( S \).
Proof
Note that \( X_\tau: \Omega \to S \) is the composition of the function \( \omega \mapsto (\omega, \tau(\omega)) \) from \( \Omega \) to \( \Omega \times T\) with the function \((\omega, t) \mapsto X_t(\omega) \) from \( \Omega \times T \) to \( S \). The first function is measurable because the two coordinate functions are measurable. The second function is measurable by assumption.
This result is one of the main reasons for the definition of a measurable process in the first place. Sometimes we literally want to stop the random process at a random time \( \tau \). As you might guess, this is the origin of the term stopping time .
Suppose again that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on the sample space \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \), and that \( \bs{X} \) is measurable. If \( \tau \) is a random time, then the process \( \bs{X}^\tau = \{X^\tau_t: t \in T\} \) defined by \( X^\tau_t = X_{t \wedge \tau} \) for \( t \in T \) is the process \( \bs{X} \) stopped at \( \tau \).
Proof
For each \( t \in T \), note that \( t \wedge \tau \) is a finite random time, and hence \( X_{t \wedge \tau} \) is measurable by the previous result. Thus \( \bs{X}^\tau \) is a well-defined stochastic process on \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \).
When the original process is progressively measurable, so is the stopped process.
Suppose again that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on the sample space \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \), and that \( \bs{X} \) is progressively measurable with respect to a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \). If \( \tau \) is a stopping time relative to \( \mathfrak{F} \), then the stopped process \( \bs{X}^\tau = \{X^\tau_t: t \in T\} \) is progressively measurable with respect to the stopped filtration \( \mathfrak{F}^\tau \).
Since \( \mathfrak{F} \) is finer than \( \mathfrak{F}^\tau \), it follows that \( \bs{X}^\tau \) is also progressively measurable with respect to \( \mathfrak{F} \).
Suppose again that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on the sample space \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \), and that \( \bs{X} \) is progressively measurable with respect to a filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) on \( (\Omega, \mathscr{F}) \). If \( \tau \) is a finite stopping time relative to \( \mathfrak{F} \) then \( X_\tau \) is measurable with respect to \( \mathscr{F}_\tau \).
For many random processes, the first time that the process enters or hits a set of states is particularly important. In the discussion that follows, let \( T_+ = \{t \in T: t \gt 0\} \), the set of positive times.
Suppose that \( \bs{X} = \{X_t: t \in T\} \) is a stochastic process on \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \). For \( A \in \mathscr{S} \), define
- \( \rho_A = \inf\{t \in T: X_t \in A\} \), the first entry time to \( A \).
- \( \tau_A = \inf\{t \in T_+: X_t \in A\} \), the first hitting time to \( A \).
As usual, \( \inf(\emptyset) = \infty \) so \(\rho_A = \infty\) if \(X_t \notin A\) for all \(t \in T\), so that the process never enters \(A\), and \( \tau_A = \infty \) if \( X_t \notin A \) for all \( t \in T_+ \), so that the process never hits \( A \). In discrete time, it's easy to see that these are stopping times.
Suppose that \( \{X_n: n \in \N\} \) is a stochastic process on \( (\Omega, \mathscr{F}) \) with state space \( (S, \mathscr{S}) \). If \( A \in \mathscr{S} \) then \(\tau_A\) and \( \rho_A \) are stopping times relative to the natural filtration \( \mathfrak{F}^0 \).
Proof
Let \( n \in \N \). Note that \(\{\rho_A \gt n\} = \{X_0 \notin A, X_1 \notin A, \ldots, X_n \notin A\} \in \sigma\{X_0, X_1, \ldots, X_n\}\). Similarly, \( \{\tau_A \gt n\} = \{X_1 \notin A, X_2 \notin A \ldots, X_n \notin A \} \subseteq \sigma\{X_0, X_1, \ldots, X_n\}\).
So of course in discrete time, \( \tau_A \) and \( \rho_A \) are stopping times relative to any filtration \( \mathfrak{F} \) to which \( \bs{X} \) is adapted. You might think that \(\tau_A\) and \( \rho_A \) should always be a stopping times, since \(\tau_A \le t\) if and only if \(X_s \in A\) for some \( s \in T_+ \) with \(s \le t\), and \( \rho_A \le t \) if and only if \( X_s \in A \) for some \( s \in T \) with \( s \le t \). It would seem that these events are known if one is allowed to observe the process up to time \(t\). The problem is that when \(T = [0, \infty)\), these are uncountable unions, so we need to make additional assumptions on the stochastic process \( \bs{X} \) or the filtration \( \mathfrak{F} \), or both.
Suppose that \( S \) has an LCCB topology, and that \( \mathscr{S} \) is the \( \sigma \)-algebra of Borel sets. Suppose also that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is right continuous and has left limits. Then \( \tau_A \) and \( \rho_A \) are stopping times relative to \( \mathfrak{F}^0_+ \) for every open \( A \in \mathscr{S} \).
Here is another result that requires less of the stochastic process \( \bs{X} \), but more of the filtration \( \mathfrak{F} \).
Suppose that \( \bs{X} = \{X_t: t \in [0, \infty)\} \) is a stochastic process on \( (\Omega, \mathscr{F}) \) that is progressively measurable relative to a complete, right-continuous filtration \( \mathfrak{F} = \{\mathscr{F}_t: t \in [0, \infty)\} \). If \( A \in \mathscr{S} \) then \( \rho_A \) and \( \tau_A \) are stopping times relative to \( \mathfrak{F} \).


15.3: Central Control
Learning Objectives
- Describe the role of higher centers of the brain in autonomic regulation
- Explain the connection of the hypothalamus to homeostasis
- Describe the regions of the CNS that link the autonomic system with emotion
- Describe the pathways important to descending control of the autonomic system
The pupillary light reflex (Figure \(\PageIndex{1}\)) begins when light hits the retina and causes a signal to travel along the optic nerve. This is visual sensation, because the afferent branch of this reflex is simply sharing the special sense pathway. Bright light hitting the retina leads to the parasympathetic response, through the oculomotor nerve, followed by the postganglionic fiber from the ciliary ganglion, which stimulates the circular fibers of the iris to contract and constrict the pupil. When light hits the retina in one eye, both pupils contract. When that light is removed, both pupils dilate again back to the resting position. When the stimulus is unilateral (presented to only one eye), the response is bilateral (both eyes). The same is not true for somatic reflexes. If you touch a hot radiator, you only pull that arm back, not both. Central control of autonomic reflexes is different than for somatic reflexes. The hypothalamus, along with other CNS locations, controls the autonomic system.
Forebrain Structures
Autonomic control is based on the visceral reflexes, composed of the afferent and efferent branches. These homeostatic mechanisms are based on the balance between the two divisions of the autonomic system, which results in tone for various organs that is based on the predominant input from the sympathetic or parasympathetic systems. Coordinating that balance requires integration that begins with forebrain structures like the hypothalamus and continues into the brain stem and spinal cord.
The Hypothalamus
The hypothalamus is the control center for many homeostatic mechanisms. It regulates both autonomic function and endocrine function. The roles it plays in the pupillary reflexes demonstrates the importance of this control center. The optic nerve projects primarily to the thalamus, which is the necessary relay to the occipital cortex for conscious visual perception. Another projection of the optic nerve, however, goes to the hypothalamus.
The hypothalamus then uses this visual system input to drive the pupillary reflexes. If the retina is activated by high levels of light, the hypothalamus stimulates the parasympathetic response. If the optic nerve message shows that low levels of light are falling on the retina, the hypothalamus activates the sympathetic response. Output from the hypothalamus follows two main tracts, the dorsal longitudinal fasciculus and the medial forebrain bundle (Figure \(\PageIndex{2}\)). Along these two tracts, the hypothalamus can influence the Eddinger–Westphal nucleus of the oculomotor complex or the lateral horns of the thoracic spinal cord.
These two tracts connect the hypothalamus with the major parasympathetic nuclei in the brain stem and the preganglionic (central) neurons of the thoracolumbar spinal cord. The hypothalamus also receives input from other areas of the forebrain through the medial forebrain bundle. The olfactory cortex, the septal nuclei of the basal forebrain, and the amygdala project into the hypothalamus through the medial forebrain bundle. These forebrain structures inform the hypothalamus about the state of the nervous system and can influence the regulatory processes of homeostasis. A good example of this is found in the amygdala, which is found beneath the cerebral cortex of the temporal lobe and plays a role in our ability to remember and feel emotions.
The Amygdala
The amygdala is a group of nuclei in the medial region of the temporal lobe that is part of the limbic lobe (Figure \(\PageIndex{3}\)). The limbic lobe includes structures that are involved in emotional responses, as well as structures that contribute to memory function. The limbic lobe has strong connections with the hypothalamus and influences the state of its activity on the basis of emotional state. For example, when you are anxious or scared, the amygdala will send signals to the hypothalamus along the medial forebrain bundle that will stimulate the sympathetic fight-or-flight response. The hypothalamus will also stimulate the release of stress hormones through its control of the endocrine system in response to amygdala input.
The Medulla
The medulla contains nuclei referred to as the cardiovascular center , which controls the smooth and cardiac muscle of the cardiovascular system through autonomic connections. When the homeostasis of the cardiovascular system shifts, such as when blood pressure changes, the coordination of the autonomic system can be accomplished within this region. Furthermore, when descending inputs from the hypothalamus stimulate this area, the sympathetic system can increase activity in the cardiovascular system, such as in response to anxiety or stress. The preganglionic sympathetic fibers that are responsible for increasing heart rate are referred to as the cardiac accelerator nerves , whereas the preganglionic sympathetic fibers responsible for constricting blood vessels compose the vasomotor nerves .
Several brain stem nuclei are important for the visceral control of major organ systems. One brain stem nucleus involved in cardiovascular function is the solitary nucleus. It receives sensory input about blood pressure and cardiac function from the glossopharyngeal and vagus nerves, and its output will activate sympathetic stimulation of the heart or blood vessels through the upper thoracic lateral horn. Another brain stem nucleus important for visceral control is the dorsal motor nucleus of the vagus nerve, which is the motor nucleus for the parasympathetic functions ascribed to the vagus nerve, including decreasing the heart rate, relaxing bronchial tubes in the lungs, and activating digestive function through the enteric nervous system. The nucleus ambiguus, which is named for its ambiguous histology, also contributes to the parasympathetic output of the vagus nerve and targets muscles in the pharynx and larynx for swallowing and speech, as well as contributing to the parasympathetic tone of the heart along with the dorsal motor nucleus of the vagus.
EVERYDAY CONNECTIONS: Exercise and the Autonomic System
In addition to its association with the fight-or-flight response and rest-and-digest functions, the autonomic system is responsible for certain everyday functions. For example, it comes into play when homeostatic mechanisms dynamically change, such as the physiological changes that accompany exercise. Getting on the treadmill and putting in a good workout will cause the heart rate to increase, breathing to be stronger and deeper, sweat glands to activate, and the digestive system to suspend activity. These are the same physiological changes associated with the fight-or-flight response, but there is nothing chasing you on that treadmill.
This is not a simple homeostatic mechanism at work because “maintaining the internal environment” would mean getting all those changes back to their set points. Instead, the sympathetic system has become active during exercise so that your body can cope with what is happening. A homeostatic mechanism is dealing with the conscious decision to push the body away from a resting state. The heart, actually, is moving away from its homeostatic set point. Without any input from the autonomic system, the heart would beat at approximately 100 bpm, and the parasympathetic system slows that down to the resting rate of approximately 70 bpm. But in the middle of a good workout, you should see your heart rate at 120–140 bpm. You could say that the body is stressed because of what you are doing to it. Homeostatic mechanisms are trying to keep blood pH in the normal range, or to keep body temperature under control, but those are in response to the choice to exercise.
Chapter Review
The autonomic system integrates sensory information and higher cognitive processes to generate output, which balances homeostatic mechanisms. The central autonomic structure is the hypothalamus, which coordinates sympathetic and parasympathetic efferent pathways to regulate activities of the organ systems of the body. The majority of hypothalamic output travels through the medial forebrain bundle and the dorsal longitudinal fasciculus to influence brain stem and spinal components of the autonomic nervous system. The medial forebrain bundle also connects the hypothalamus with higher centers of the limbic system where emotion can influence visceral responses. The amygdala is a structure within the limbic system that influences the hypothalamus in the regulation of the autonomic system, as well as the endocrine system.
These higher centers have descending control of the autonomic system through brain stem centers, primarily in the medulla, such as the cardiovascular center. This collection of medullary nuclei regulates cardiac function, as well as blood pressure. Sensory input from the heart, aorta, and carotid sinuses project to these regions of the medulla. The solitary nucleus increases sympathetic tone of the cardiovascular system through the cardiac accelerator and vasomotor nerves. The nucleus ambiguus and the dorsal motor nucleus both contribute fibers to the vagus nerve, which exerts parasympathetic control of the heart by decreasing heart rate.
Interactive Link Questions
Watch this video to learn about physical responses to emotion. The autonomic system, which is important for regulating the homeostasis of the organ systems, is also responsible for our physiological responses to emotions such as fear. The video summarizes the extent of the body’s reactions and describes several effects of the autonomic system in response to fear. On the basis of what you have already studied about autonomic function, which effect would you expect to be associated with parasympathetic, rather than sympathetic, activity?
Answer: The release of urine in extreme fear. The sympathetic system normally constricts sphincters such as that of the urethra.
Review Questions
Q. Which of these locations in the forebrain is the master control center for homeostasis through the autonomic and endocrine systems?
A. hypothalamus
B. thalamus
C. amygdala
D. cerebral cortex
Answer: A
Q. Which nerve projects to the hypothalamus to indicate the level of light stimuli in the retina?
A. glossopharyngeal
B. oculomotor
C. optic
D. vagus
Answer: C
Q. What region of the limbic lobe is responsible for generating stress responses via the hypothalamus?
A. hippocampus
B. amygdala
C. mammillary bodies
D. prefrontal cortex
Answer: B
Q. What is another name for the preganglionic sympathetic fibers that project to the heart?
A. solitary tract
B. vasomotor nerve
C. vagus nerve
D. cardiac accelerator nerve
Answer: D
Q. What central fiber tract connects forebrain and brain stem structures with the hypothalamus?
A. cardiac accelerator nerve
B. medial forebrain bundle
C. dorsal longitudinal fasciculus
D. corticospinal tract
Answer: B
Critical Thinking Questions
Q. Horner’s syndrome is a condition that presents with changes in one eye, such as pupillary constriction and dropping of eyelids, as well as decreased sweating in the face. Why could a tumor in the thoracic cavity have an effect on these autonomic functions?
A. Pupillary dilation and sweating, two functions lost in Horner’s syndrome, are caused by the sympathetic system. A tumor in the thoracic cavity may interrupt the output of the thoracic ganglia that project to the head and face.
Q. The cardiovascular center is responsible for regulating the heart and blood vessels through homeostatic mechanisms. What tone does each component of the cardiovascular system have? What connections does the cardiovascular center invoke to keep these two systems in their resting tone?
A. The heart—based on the resting heart rate—is under parasympathetic tone, and the blood vessels—based on the lack of parasympathetic input—are under sympathetic tone. The vagus nerve contributes to the lowered resting heart rate, whereas the vasomotor nerves maintain the slight constriction of systemic blood vessels.
Glossary
- cardiac accelerator nerves
- preganglionic sympathetic fibers that cause the heart rate to increase when the cardiovascular center in the medulla initiates a signal
- cardiovascular center
- region in the medulla that controls the cardiovascular system through cardiac accelerator nerves and vasomotor nerves, which are components of the sympathetic division of the autonomic nervous system
- dorsal longitudinal fasciculus
- major output pathway of the hypothalamus that descends through the gray matter of the brain stem and into the spinal cord
- limbic lobe
- structures arranged around the edges of the cerebrum that are involved in memory and emotion
- medial forebrain bundle
- fiber pathway that extends anteriorly into the basal forebrain, passes through the hypothalamus, and extends into the brain stem and spinal cord
- vasomotor nerves
- preganglionic sympathetic fibers that cause the constriction of blood vessels in response to signals from the cardiovascular center
Contributors and Attributions
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OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


1.2: Becoming a Skilled Observer
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- Gina Peterson and Emily Elam
- College of the Canyons
To truly observe a child, you must be present, knowledgeable, inquisitive and intentional. With every observation, you will sharpen your skills as you learn how to effectively gather objective evidence and detailed data.
Be present : To capture all the individual mannerisms, subtle social nuances, non-verbal body language and dynamic conversations that occur throughout the day you must be attentive, focused and ready to go at any given moment. Children move fast. When we blink, we are bound to miss some little detail or precious moment, that’s a given. Being present takes considerable effort and careful planning.
Be knowledgeable : Understanding the core concepts of early childhood education is extremely important if you are to set reasonable expectations and plan developmentally appropriate learning experiences. Familiarizing yourself with child development theories will help you understand and appreciate why children do what they do. Learning about the key principles in early care and education will provide you with a solid foundation and a wide range of instructional strategies to support a child’s development.
Be inquisitive : Think of yourself as a researcher. Your primary mission is to investigate the children in your care by routinely gathering evidence, using a variety of observation methods and tools. As a good researcher you will need to ask some thoughtful questions. These questions will guide you as you plan purposeful observations and as you select your method of observation. Here are some sample questions you may ask yourself: What activities interest Max? How many times did Stevie hit today? What skills did Hazel master today with this activity, and what skills need further support? How long did Zoey stay engaged while playing in the sandbox? What milestones will this activity support? By a sking thoughtful questions, you will learn more about the children in your care and you will do a better job at supporting each child’s individual needs. Rather than fixating on a child’s behavior, in time you will begin using focused observations to try and figure out the reasons why a child acts the way they do.
Be intentional : As you organize learning experiences, set up the classroom and outside environment, assess children’s developmental progress, engage in activities, and interact with your children and families – you must have a thoughtful plan of action in place. “Intentional teaching means that everything you do as a teacher has a specific goal and purpose” (Gordon & Browne, 2016 p. 103). Even as spontaneous situations arise, intentional teachers must make the most of teachable moments. Intentional teachers conduct regular observations and gather objective documentation data to be accountable for the actions they take, the plans they generate and the assessments they make.