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Apr 23, 2022 · 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.
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 $X_1, X_2, . . . , X_n$.
6.2 Stopping Times Let (Ω,F,P) be a probability space and let (F t) t⩾0 be a filtration of (Ω,F,P). 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 ...
Apr 23, 2022 · Let \( \mathfrak F = \{\mathscr{F}_k: k \in \N_n^+\} \) be the natural filtration of \( \bs X \), and suppose that \( \rho \) is a stopping time for \( \mathfrak F \). Define \( \bs Y = \{Y_k: k \in \N_n\} \) by \( Y_0 = 0 \) and \( Y_k = X_{\rho \wedge k} \vee a_{n-k} \) for \( k \in \N_n^+ \).
Feb 10, 2018 · The first time that an adapted process X t hits a given value or set of values is a stopping time. The inclusion of ∞ into the range of τ is to cover the case where X t never hits the given values.
- StoppingTime
- 2013-03-22 14:41:13
- 2013-03-22 14:41:13
- stopping time
Stopping time is a random variable τ with values in [0, ≥ ∞] and such that {τ ≤ t} ∈ Ft for t ≥ 0. We can think of stopping time τ as a strategy which at every time t decides to stop or not (until stopping decision is made) on base of the information available so far.
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The event f. tg means you have taken action on or before time t. The event being 2 F(t) then means your decision of taking action on or before time t entirely depends on the information up to time t, i.e. it does not involve future information. Such is called a stopping time and it is an important concept to study.
