Let (Ω,F,P) be a probability space. Let (Fn)n∈N be a filtration. Let T:Ω→N∪{+∞}be a random time. T is called a stopping time with respect to Fn if ∀n∈N:{T≤n}∈Fn
T:Ω→R+∪{+∞} is a stopping time if {T≤t}∈Ft ∀t∈R+
Motivation
We usually want to take an action when a MC satisfies certain properties. The stopping time describes those strategies are realizable in reality. Any deterministic time N≥0 is trivially a stopping time, since {N=n}={∅ \mboxifn=NΩ \mboxifn=N ## Remark The hitting time TA of a subset A is a stopping time with Bn=Ac∗⋯∗Ac∗Awith the complements, Ac, comprising n terms.
Example
Any realistic decision takes place at a time which is a stopping time. Consider an optimal investment problem: if an investor claims to stop investing (e.g., purchasing houses) when the investment (value of the housing market) is at its local peak, the decision instant could not be a stopping-time in general: this peak-time is not a stopping time because to find out whether the investment value is at its peak, the next realization should be known, and this information is not available up to any given time in a causal fashion for a non-trivial (i.e., non-deterministic) stochastic process.
Let (Fn)n∈N be a filtration on (Ω,F,P), and let T be a stopping time. We define FT={A∈F:A∩{T≤n}∈Fn,∀n∈N} FT is the “σ-algebra of events up to stopping time T”.
Let T be a (Ft)t≥0-stopping time. We define FT={A∈F:A∩{T≤t}∈Ft, ∀t≥0}
Let S,T be (Fn)n∈N-stopping times in (Ω,F,P) then 1. Monotonicity: S≤T⟹FS⊂FT 2. If T(ω)=N,∀ω∈Ω,N∈N then FT=FN 3. min(S,T),max(S,T) are also stopping times.
Let S,T be (Fn)n∈N-stopping times in (Ω,F,P) then 4. Monotonicity: S≤T⟹FS⊂FT 5. min(S,T),max(S,T) are also stopping times. 6. Let (Sn)n∈N be an increasing sequence of stopping times. Then n∈NsupSn is a (Ft)t≥0-stopping time 7. Let (Sn)n∈N be an decreasing sequence of stopping times. Then n∈NinfSn is a (Ft+)t≥0-stopping timewhere t+ is the right limit and Ft+=s>t⋂Fs