stopping time properties

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 \( (\Omega, \mathscr{F}) \). Hence \( \{\tau \lt t_n\} \in \mathscr{F}_{t_n} \subseteq \mathscr{F}_s \) for \( n \ge k \), and so it follows that \( \{\tau \le t\} \in \mathscr{F}_s \). If \( A \in \mathscr{S} \) then \( \rho_A \) and \( \tau_A \) are stopping times relative to \( \mathfrak{F} \). For the remainder of this section, we have a fixed A family of \( \sigma \)-algebras \(\mathfrak{F} = \{\mathscr{F}_t: t \in T\} \) is a So a filtration is simply an increasing family of sub-\(\sigma\)-algebras of \( \mathscr{F} \), indexed by \( T \). 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. Then each of the following events is in \(\mathscr{F}_\tau\) and in \(\mathscr{F}_\rho\).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 \( \mathfrak{F} \). 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 \( (\Omega, \mathscr{F}) \). \[ \mathscr{F}_{t++} = \bigcap\{\mathscr{F}_{s+}: s \in (t, \infty)\} = \bigcap\left\{\bigcap\{\mathscr{F}_r: r \in (s, \infty)\}: s \in (t, \infty)\right\} = \bigcap\{\mathscr{F}_u: u \in (t, \infty)\} = \mathscr{F}_{t+} \] \( \newcommand{\var}{\text{var}} \) 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. (b) A player must stop if he wins all of this money or goes into debt by this amount. 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. If this service is disabled, any services that explicitly depend on it will fail to start. ThenNote that when \(T = \N\), we actually showed that \(\{\tau \lt t\} \in \mathscr{F}_{t-1}\) and \(\{\tau \ge t\} \in \mathscr{F}_{t-1}\). 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 Clearly if \( \bs{X} \) is progressively measurable with respect to a filtration, then it is progressively measurable with respect to any finer filtration. \( \newcommand{\cov}{\text{cov}} \) 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 \( t \). 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 \( (\Omega, \mathscr{F}) \). Hence First, if X is a process and τ is a stopping time, then X is used to denote the process X stopped at time τ. 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. But this is not obvious, and in fact is not true without additional assumptions. \[ \sigma(\mathscr{F}_s \cup \mathscr{N}) \subseteq \sigma(\mathscr{F}_t \cup \mathscr{N}) \subseteq \sigma(\mathscr{F} \cup \mathscr{N})\]
Then \(\tau\) is measureable with respect to \(\mathscr{F}_\tau\).It suffices to show that \(\{\tau \le s\} \in \mathscr{F}_\tau\) for each \(s \in T\). If \( \tau \) is a random time, we are often interested in the state \( X_\tau \) at the random time. A random time \( \tau \) is a In a sense, a stopping time is a random time that does not require that we see into the future. 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 \(t \in T\).

The first function is measurable because the two coordinate functions are measurable. 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. 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 \( A \cap \{\tau \le r\} \in \mathscr{F}_r \). 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 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. For \( t \in T \) define \( \mathscr{F}^\tau_t = \mathscr{F}_{t \wedge \tau} \). 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 \( \sigma\left(\bigcup_{i \in I} \mathscr{F}^i_s\right) \subseteq \sigma\left(\bigcup_{i \in I} \mathscr{F}^i_t\right) \subseteq \mathscr{F} \).Note again that we can have a filtration without an underlying stochastic process in the background.
That is, we can tell whether or not \( \tau \le t \) from our information at time \( t \).

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