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In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time [1]) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest.
May 18, 2015 · The definition I am provided with is as follows: A random time $τ$ is called a stopping time if for any $n$, one can decide whether the event $\ {τ ≤ n\}$ (and hence the complementary event $\ {τ > n\}$) has occurred by observing the first n variables $X_1, X_2, . . . , X_n$.
Apr 23, 2022 · 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 \( t \in T \).
Definition 6.8.A random variable T : Ω →[0,∞] is called a stopping time w.r.t. the filtration (F t) t⩾0 if {T⩽t}∈F t for all t⩾0. Consequently, if Tis a (F t) t⩾0 stopping time, then {T<t}= [∞ n=1 ˆ T⩽t− 1 n ˙ ∈F t and {T= t}∈F t. Lemma 6.9. Let G t = F t+,t⩾0. Then {T <t}∈F t, for all t⩾0, iff T is a(G t) t⩾0 ...
A random variable \( \tau \) taking values in \( T_\infty \) is called a random time. Suppose that \( \mf F = \{\ms F_t: t \in T\} \) is a filtration on \( (\Omega, \ms F) \). A random time \( \tau \) is a stopping time relative to \( \mf F \) if \( \{\tau \le t\} \in \ms F_t \) for each \( t \in T \).
Apr 23, 2022 · Recall that a random time \( \tau \) with values in \( T \cup \{\infty\} \) is a stopping time relative to \( \mathfrak F \) if \( \{\tau \le t\} \in \mathscr{F}_t \) for \( t \in T \). So a stopping time is a random time that does not require that we see into the future.
Oct 11, 2017 · Stopping time. Let Ft, t ∈ T, be a non-decreasing family of sub- σ -algebras on a measurable space (Ω, F), where T is an interval in [0, ∞] or a subset of {0, 1, 2, …, ∞}. Then a stopping time (relative to this family of subalgebras) is a mapping (a random variable) τ: Ω → T ∪ {∞} such that {τ(ω) ≤ t} ∈ Ft for all t ∈ T.
