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Stopping times and sigma-algebra of tau-history

posted by Nikolay Baldin on August 8, 2015

This post is on quite a well-presented topic in stochastic processes that was opened to me from a different perspective recently.

Below I am talking about stopping times, sigma-algebra of \(\tau\)-history \(\mathcal{F}_\tau\) for time-indexed stochastic processes (discrete and continuous time) and its properties. In next posts, I'm planing to give some extensions of the results below to set-indexed stochastic processes (with no direct order).

This material might be interesting, in particular, for students studying stochastic processes. My main motivation is to discuss a few quite important questions below that often remain hidden in this topic.

0. Introducing stopping times and sigma-algebras

Stopping times is quite a well-studied topic in graduate courses in stochastic processes and can be found in almost every book in stochastic processes. In this post, I follow mostly O. Kallenberg ''Foundations of Modern Probability'' and I. Karatzas, S. Shreve ''Brownian Motion and Stochastic Calculus''.

Let us fix a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) and an index set \(T \subset \mathbb{R}\) with its sigma-algebra \(\mathcal{B}_T\). A filtration on \(\mathcal{F}\) is defined as a decresing sequence of \(\sigma\)-algebra \(\mathcal{F}_t\) for \(t\in T\). A process \(X\) is said to be adapted to the filtration \(\{\mathcal{F}_t\}_{t\in T}\) if \(X_t\) is measurable w.r.t. \(\mathcal{F}_t\) for all \(t\in T \). One can show that the smallest sigma-algebra such that the process is adapted to it is a natural or canonical filtration \(\F_t = \sigma(X_s, s \le t) \). A random variable \(\tau : (\Omega, \mathcal{F}) \to (T, \mathcal{B}_T)\) is called a stopping time if \(\{\tau \le t \} \in \mathcal{F}_t\). With a stopping time \(\tau\), we may associate a set $$\F_\tau = \{A \in \F: A\cap \{\tau \le t\} \in \F_t, \forall t \in T\}.\nonumber$$ One can easily show that \(\F_\tau\) as defined above, is actually a sigma-algebra.

It is often reasonable to distinguish between stopping times and optional times (a random time is optional if \(\{\tau < t\} \in \F_t\)). Different authors use different notions of this time and it's important not to mix it up.

  • \(\{\tau \le t\} \in \F_t\)
  • stopping time (in Kallenberg and Karatzas, Shreve), optional time (in Kallenberg), Markov time (Shiryaev)
  • \(\{\tau < t\} \in \F_t\)
  • weakly optional (in Kallenberg) and optional time (in Karatzas, Shreve)

    Note, that A. Shiryaev, "Optimal stopping rules" calls a Markov time \(\tau \) a stopping time, only if \(\tau\) is finite almost surely, i.e. \(\P(\tau < \infty) = 1\), and Kallenberg does not distinguish these situations in his definition. What's so special about the case \(\P(\tau = \infty) > 0\)? For the results discussed below, it does not matter, but many other results including Optional Sampling theorem, which are beyond the scope of these notes, require the stopping time to be finite.

    1. Basic properties

    For stopping times, we have
    1. A constant \(c \in T \) is a stopping time.
    2. If \(\sigma, \tau\) are stopping times, then also \(\sigma \vee \tau \), \(\sigma \wedge \tau \) and \(\sigma + \tau \) are stopping times. However, \(\sigma - \tau \) is not a stopping time!
    3. For a stopping time \(\tau\), we have \( \{ \tau < t\},\{ \tau > t\}, \{ \tau = t\} \) are in \(\F_\tau \) for all \(t \in T \). In particular \(\tau \) is \(\F_\tau \)-measurable.

    Note, from 3. follows that \(\sigma(\tau) \subseteq \F_\tau \). All these facts are easily proved using approximation arguments. It's quite evident that \(\F_\tau \nsubseteq \sigma(\tau) \) in general. A non-trivial question: what would be an example of a stopping time \(\tau \) such that \(\F_\tau = \sigma(\tau) \)?

    For sigma-algebras, we have

    1. A very desired property for understanding \(\F_\tau \), that \(\F_\tau = \F_t \) on \(\{\tau = t\} \) for all \(t\in T \).
    2. An intuitively clear property, that \(\F_\sigma \cap \{\sigma \le \tau\} \subseteq \F_{\sigma\wedge\tau} = \F_\sigma \cap \F_\tau \) for any stopping time \(\tau \) and \(\sigma \).
    Note, 5. implies that for stopping sets \(\tau \) and \(\sigma \) such that \(\tau \le \sigma\), we have \(\F_\tau \subseteq \F_\sigma \). The proofs can be found in Kallenberg.

    2. More advanced results

    Advanced results here are somehow related to stopped processes. Recall, with a stochastic process \(X \) and stopping time \(\tau \), we may associate a stopped process \(X^{\tau} \) defined as \(X^{\tau}_t(\omega) = X_{\tau(\omega)\wedge t} (\omega)\), which ''stops'' after time \(\tau \). For proving certain results about the stopped process, the following representation is very useful $$X^\tau_t(\omega) = X_t {\bf 1}(t \le \tau) + X_\tau {\bf 1}(t> \tau)\nonumber$$ See also next figure for an illustration.

    What can we say about this process? Let \(X_\tau \) denote the value of the process \(X \) at the time \(\tau \) (one can show that \(X_\tau \) is a random variable). We also have

    1. For a stopping time \(\tau\), the random variable \(X_{\tau} \) is \(\F_\tau \)-measurable. Consequently, \(X^\tau_t(\omega) \) is measurable with respect to \(\F_{\tau \wedge t} \) and so \(\F_t \).
    The proof is quite straightforward in discrete time. In continuous time, it holds when the process is progressively measurable. Even more is true: if \(X \) is a martingale with respect to the filtration \(\F_t \), then so is \(X^\tau \).

    A non-trivial question: what's the generated filtration of this process \(\sigma(X^\tau_s(\omega),s \le t) \) and how is it related to the filtration \(\F_t \)? It's quite desirable to have $$\F_{\tau\wedge t} = \sigma(X^\tau_s(\omega),s \le t) \label{ref1}$$ isn't it? This statement holds under a specific assumption on the probability space. The proof, especially the part \(\F_{\tau\wedge t} \subseteq \sigma(X^\tau_s(\omega),s \le t) \), is non trivial, but I will comment on it later. Thus, we have a nice representation of the stopped sigma algebra $$\F_\tau = \sigma(X^\tau_s(\omega),s >0 ) \label{ref2}$$ The inclusion \(\sigma(X^\tau_s(\omega),s >0 ) \subseteq \F_\tau \) is quite evident: we use that \(X_{\tau \wedge s} \) is \(\F_{\tau\wedge s} \)-measurable using Property 6. from above and \(\F_{\tau\wedge s} \subseteq \F_\tau\). The proof of the other inclusion is more involved. One may follow the proof of Lemma 1.3.3 in D. Stroock, S. Varadhan "Multidimensional Diffussion Processes", OR one can try to derive it from \ref{ref1}.

    Note, that we can actually derive \ref{ref1} from \ref{ref2} using that \(\tau \wedge t \) is a stopping time and so \( \F_{\tau \wedge t} = \sigma(X_s^{\tau\wedge t},s >0) \).

    3.1 Arbitrary filtration \(\F \)

    In this discussion, we have assumed the filtration is induced by the process. One can also consider an arbitrary filtration \(\F \), define a stopping time \(\tau\) and the sigma-albegra of \(\tau\)-history \(\F_\tau \). Then the following representation holds, $$\F_\tau = \sigma(X_\tau; X \,\,\text{all adapted, cadlag processes} ).$$ The proof can be found in p.6 in P. Protter ''Stochastic integration and differential equations''.

    4. Discrete vs Continuous case. Some proofs

    Here, I would like to sketch some of the proofs of properties above in the discrete and continuous time scenarios separately (so far we have not really specified the set \(T \)). In discrete time, I will assume the number of elements in \(T \) to be at most countable. Most of the statements hold in both cases, but the proofs are usually easier in discrete times.

    4.1 Discrete-time case.

    A standard discrete-time case is \(T = \mathbb{Z}_{+} \cup \{\infty\} \). In this case, all the results of section 1. hold. The proofs are quite simple and rely on the fact that the set \(T \) consists of countable number of elements. For example, to prove 6. in Section 2, i.e. that \(X_\tau \) is \(\F_\tau\)-measurable, we use \( \{ X_\tau \in A \} \cap \{ \tau \le t \} = \cup_{ s \in T, s \le t} ( \{\tau = s \} \cap \{ X_\tau \in A\} ) = \cup_{ s \in T, s \le t} ( \{\tau = s \} \cap \{ X_s \in A\} ) \in \F_t \) for all \(t \in T \).

    It turns out, that not all the results of Section 2 hold in the discrete-time case. In general, we don't have neither \ref{ref1} nor \ref{ref2}. Instead, one can show that $$\F_\tau = \sigma(X^\tau_{n},\tau, n \ge 0). \nonumber$$ To prove \(\subseteq \) inclusion, one can use \(A = \cup_{n} (A \cap \{\tau = n\} )\) for all \(A \in \F_{\tau} \). For the other inclusion, one can show directly that the generator of the sigma-algebra \(\sigma(X^\tau_{n},\tau, n \ge 0) \) is in \(\F_\tau \). Unfortunately, in general we have $$\sigma(X_{n\wedge\tau},\tau, n \ge 0) \neq \sigma(X_{n\wedge\tau}, n \ge 0)\label{ref3}$$ and so \(\F_\tau \neq \sigma(X_{n\wedge\tau}, n \ge 0) \). In particular, for all \(k \in \N \) we have $$ \F_{\tau\wedge k} = \sigma(X_{n\wedge(\tau\wedge k)},\tau \wedge k, n \ge 0) = \sigma(X_{\tau \wedge n},\tau \wedge k, n \le k).\nonumber$$

    A non-trivial question: an example for \(\F_\tau \neq \sigma(X_{n\wedge\tau}, n \ge 0) \)? First, one may think that the set \(\{\tau = k \} , k\in \N \) can not be described in terms of the paths of the stopped process, i.e. there might be a path that stops at the moment \(\tau = k + 1 \) and has the same value at time \(k \), i.e. \(X_k = X_{k+1} \). But this situation is impossible and one can prove that if for \(\omega \in \Omega \) we have \(\tau(\omega) \le t \) and there is \( \omega^\prime\) such that \(X_s(\omega) = X_s(\omega^\prime) \) for all \(s \le t \), then necessarily \(\tau(\omega) = \tau(\omega^\prime) \).

    A key ingredient that assures \ref{ref3} and thus \ref{ref1} and \ref{ref2} is that the probability space is ''rich'' enough, in the sense that for each \(t \in T \) and each \(\omega \in \Omega \) there will be \(\omega^\prime \in \Omega \) such that $$X_s(\omega^\prime) = X_{s\wedge t}(\omega) \nonumber$$ for all \(s \in T \). For the proof, see Theorem 6 in Shiryaev, "Optimal stopping rules" or Lemma 1.3.3 in D. Stroock, S. Varadhan "Multidimensional Diffussion Processes"

    4.2 Continuous-time case.

    Now, consider the continuous case \(T = [0,\infty] \). Again, all the results of Section 1. hold. Most of the statements of Section 2. require somehow either the process \(X \) to be at least right-continuous with left limits or the filtration to be right-continuous. In particular, Property 6 holds when the process is progressively measurable, which is implied by right-continuity, see Proposition 2.18 and 1.13 in Karatzas, Shreve. Note, that when the filtration is right-continuous, every optional time is also a stopping time.

    Recall: in general, the continuity of the process does not imply the continuity of its natural filtration. A non-trivial question: can you construct a right-continuous process, whose natural filtration is not right-continuous?

    When the filtration is right-continuous, we can derive all the properties of the stopped process identical to those in discrete time using discretization of a stopping time. For this, we need the following result:

    1. If \(\tau_n \downarrow \tau \) are stopping times and the filtration \(F\) is right-continuous, that \(\F_\tau = \cap_n \F_{\tau_n} \).
    Thus, we can derive $$\F_\tau = \sigma(X^\tau_{t},\tau, t \ge 0). \nonumber$$ As I mentioned, so much desired property, \(\F_\tau = \sigma(X^\tau_{t}, t \ge 0) \), holds under the assumption that the probability space is ''rich'' enough, see above.