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) 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. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time. Stopping times occur in decision theory, and the optional stopping theorem is an important result in this context. Stopping times are also frequently applied in mathematical proofs to "tame the continuum of time", as Chung put it in his book (1982).
Definition
Discrete time Let τ {\displaystyle \tau } be a random variable, which is defined on the filtered probability space ( Ω , F , ( F n ) n ∈ N , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },P)} with values in N ∪ { + ∞ } {\displaystyle \mathbb {N} \cup \{+\infty \}} . Then τ {\displaystyle \tau } is called a stopping time (with respect to the filtration F = ( ( F n ) n ∈ N ) {\displaystyle \mathbb {F} =(({\mathcal {F}}_{n})_{n\in \mathbb {N} })} ) if the following condition holds:
{ τ = n } ∈ F n {\displaystyle \{\tau =n\}\in {\mathcal {F}}_{n}} for all n {\displaystyle n}
Intuitively, this condition means that the "decision" of whether to stop at time n {\displaystyle n} must be based only on the information present at time n {\displaystyle n} , not on any future information.
General case Let τ {\displaystyle \tau } be a random variable, which is defined on the filtered probability space ( Ω , F , ( F t ) t ∈ T , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)} with values in T {\displaystyle T} . In most cases, T = [ 0 , + ∞ ) {\displaystyle T=[0,+\infty )} . Then τ {\displaystyle \tau } is called a stopping time (with respect to the filtration F = ( F t ) t ∈ T {\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}} ), if the following condition holds:
{ τ ≤ t } ∈ F t {\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t}} for all t ∈ T {\displaystyle t\in T}
As adapted process Let τ {\displaystyle \tau } be a random variable, which is defined on the filtered probability space ( Ω , F , ( F t ) t ∈ T , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)} with values in T {\displaystyle T} . Then τ {\displaystyle \tau } is called a stopping time if the stochastic process X = ( X t ) t ∈ T {\displaystyle X=(X_{t})_{t\in T}} , defined by
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