hitting times are stopping times


Let (ℱt)t∈𝕋 be a filtrationMathworldPlanetmathPlanetmath (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable spaceMathworldPlanetmathPlanetmath (Ω,ℱ). If X is an adapted stochastic processMathworldPlanetmath taking values in a measurable space (E,𝒜) then the hitting timePlanetmathPlanetmath of a set S∈𝒜 is defined as

τ:Ω→𝕋∪{±∞},
τ⁢(ω)=inf⁡{t∈𝕋:Xt⁢(ω)∈S}.

We suppose that 𝕋 is a closed subset of ℝ, so the hitting time τ will indeed lie in 𝕋 whenever it is finite. The main cases are discrete-time when 𝕋=ℤ+ and continuous-time where 𝕋=ℝ+. An important property of hitting times is that they are stopping times, as stated below for the different cases.

Discrete-time processes

For discrete-time processes, hitting times are easily shown to be stopping times.

Theorem.

If the index setMathworldPlanetmathPlanetmath T is discrete, then the hitting time τ is a stopping time.

Proof.

For any s≤t∈𝕋 then Xs will be ℱt/𝒜-measurable, as it is adapted. So, by the fact that the σ-algebra ℱt is closed under taking countableMathworldPlanetmath unions,

{τ≤t}=⋃s∈𝕋,s≤tXs-1(S)∈ℱt

as required. ∎

Continuous processes

For continuous-time processes it is not necessarily true that a hitting time is even measurable, unless further conditions are imposed. Processes with continuousMathworldPlanetmathPlanetmath sample paths can be dealt with easily.

Theorem.

Suppose that X is a continuous and adapted process taking values in a metric space E. Then, the hitting time τ of any closed subset S⊆E is a stopping time.

Proof.

We may suppose that S is nonempty, and define the continuous function dS⁢(x)≡inf⁡{d⁢(x,y):y∈S} on E. Then, τ is the first time at which Yt≡dS⁢(Xt) hits 0. Letting U be any countable and dense subsetPlanetmathPlanetmath of 𝕋∩[0,t] then the continuity of the sample paths of Y gives,

{τ≤t}={infu∈UYu=0}.

As the infimumMathworldPlanetmath of a countable set of measurable functionsMathworldPlanetmath is measurable, this shows that {τ≤t} is in ℱt. ∎

Right-continuous processes

Right-continuous processes are more difficult to handle than either the discrete-time and continuous sample path situations. The first time at which a right-continuous process hits a given value need not be measurable. However, it can be shown to be universally measurable, and the following result holds.

Theorem.

Suppose that X is a right-continuous and adapted process taking values in a metric space E, and that the filtration (Ft) is universally complete. Then, the hitting time τ of any closed subset S⊆E is a stopping time.

In particular, the hitting time of any closed set S⊆ℝ for an adapted right-continuous and real-valued process is a stopping time.

The proof of this result is rather more involved than the case for continuous processes, and the condition that ℱt is universally complete is necessary.

Progressively measurable processes

The début D⁢(A) of a set A⊆𝕋×Ω is defined to be the hitting time of {1} for the process 1A,

D⁢(A)⁢(ω)=inf⁡{t∈𝕋:(t,ω)∈A}.

An important result for continuous-time stochastic processes is the début theorem.

Theorem (Début theorem).

Suppose that the filtration (Ft) is right-continuous and universally complete. Then, the début D⁢(A) of a progressively measurable A⊆T×Ω is a stopping time.

Proofs of this typically rely upon properties of analytic setsMathworldPlanetmath, and are therefore much more complicated than the result above for right-continuous processes.

A process X taking values in a measurable space (E,𝒜) is said to be progressive if the set X-1⁢(S) is progressively measurable for every S∈𝒜. In particular, the hitting time of S is equal to the début of X-1⁢(S) and the début theorem has the following immediate corollary.

Theorem.

Suppose that the filtration (Ft) is right-continuous and universally complete, and that X is a progressive process taking values in a measurable space (E,A). Then, the hitting time τ of any set S∈A is a stopping time.

Title hitting times are stopping times
Canonical name HittingTimesAreStoppingTimes
Date of creation 2013-03-22 18:39:06
Last modified on 2013-03-22 18:39:06
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 60G40
Classification msc 60G05
Defines hitting time
Defines début
Defines debut
Defines début theorem
Defines debut theorem