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Created by Knut M. Synstadfrom the Noun Project

Stopping Time

Definition (Stopping Time)

Let (Ω,F,P)(\Omega,\mathcal{F},P) be a probability space. Let (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}} be a filtration. Let T:Ω→N∪{+∞}T:\Omega\to \mathbb{N}\cup \{ +\infty \}be a random time. TT is called a stopping time with respect to Fn\mathcal{F}_{n} if ∀n∈N:{T≤n}∈Fn\forall n\in\mathbb{N}:\{ T\le n \}\in\mathcal{F}_{n}

Definition (Stopping Time)

T:Ω→R+∪{+∞}T:\Omega\to \mathbb{R}^{+}\cup \{ +\infty \} is a stopping time if {T≤t}∈Ft  ∀t∈R+\{ T\le t \}\in\mathcal{F}_{t} \ \ \forall t\in\mathbb{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≥0N\ge0 is trivially a stopping time, since {N=n}={∅ \mboxifn≠NΩ \mboxifn=N\{N=n\}=\left\{\begin{align*} \emptyset \ \mbox{ if }n\not=N\\ \Omega \ \mbox{ if }n=N \end{align*}\right.

Remark

The hitting time TAT^{A} of a subset AA is a stopping time with Bn=Ac∗⋯∗Ac∗AB_{n}=A^{c}*\cdots* A^{c}*Awith the complements, AcA^{c}, comprising nn 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.

Definition

Let (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}} be a filtration on (Ω,F,P)(\Omega,\mathcal{F},P), and let TT be a stopping time. We define FT={A∈F:A∩{T≤n}∈Fn,∀n∈N}\mathcal{F}_{T}=\{ A\in\mathcal{F}:A\cap \{ T\le n\}\in\mathcal{F}_{n},\forall n\in\mathbb{N} \} FT\mathcal{F}_{T} is the “σ-algebra of events up to stopping time TT”.

Definition (σ\sigma-algebra of events up to stopping time — Continuous)

Let TT be a (Ft)t≥0(\mathcal{F}_{t})_{t\ge 0}-stopping time. We define FT={A∈F:A∩{T≤t}∈Ft, ∀t≥0}\mathcal{F}_{T}=\{ A\in\mathcal{F}:A\cap \{ T\le t \}\in\mathcal{F}_{t}, \ \forall t\ge 0 \}

Proposition (Relationship of Two Stopping Times — Discrete)

Let S,TS,T be (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}}-stopping times in (Ω,F,P)(\Omega,\mathcal{F},P) then

  1. Monotonicity: S≤T  ⟹  FS⊂FTS\le T\implies \mathcal{F}_{S}\subset \mathcal{F}_{T}
  2. If T(ω)=N,∀ω∈Ω,N∈NT(\omega)=N,\forall\omega\in\Omega, N\in\mathbb{N} then FT=FN\mathcal{F}_{T}=\mathcal{F}_{N}
  3. min(S,T),max(S,T)min(S,T),max(S,T) are also stopping times.

Proposition (Relationship of Two Stopping Times — Continuous)

Let S,TS,T be (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}}-stopping times in (Ω,F,P)(\Omega,\mathcal{F},P) then

  1. Monotonicity: S≤T  ⟹  FS⊂FTS\le T\implies \mathcal{F}_{S}\subset \mathcal{F}_{T}
  2. min(S,T),max(S,T)min(S,T),max(S,T) are also stopping times.
  3. Let (Sn)n∈N(S_{n})_{n\in\mathbb{N}} be an increasing sequence of stopping times. Then sup⁡n∈NSn is a (Ft)t≥0-stopping time\sup_{n\in\mathbb{N}}S_{n}\text{ is a }(\mathcal{F}_{t})_{t\ge 0}\text{-stopping time}
  4. Let (Sn)n∈N(S_{n})_{n\in\mathbb{N}} be an decreasing sequence of stopping times. Then inf⁡n∈NSn is a (Ft+)t≥0-stopping time\inf_{n\in\mathbb{N}}S_{n}\text{ is a }(\mathcal{F}_{t^{+}})_{t\ge 0}\text{-stopping time}where t+t^{+} is the right limit and Ft+=⋂s>tFs\mathcal{F_{t^{+}}}=\bigcap_{s>t}\mathcal{F}_{s}

Linked from

Stochastic Interval

Stopping Time Integral

Markov Property

Doob's Optional Sampling Theorem

Local Martingale

Martingale Equivalence for Stopping Times

Stopped Process is also R.C. Martingale

Passage Time

Random Time

Stopping Time