hitting times are stopping times

Let ${({ℱ}_t)}_{t\in 𝕋}$ be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space $(\mathrm{\Omega },ℱ)$. If $X$ is an adapted stochastic process taking values in a measurable space $(E,𝒜)$ then the hitting time of a set $S\in 𝒜$ is defined as

$\tau :\mathrm{\Omega }\to 𝕋\cup \{\pm \mathrm{\infty }\},$

$\tau (\omega )=inf\{t\in 𝕋:X_t(\omega )\in S\}.$

We suppose that $𝕋$ is a closed subset of $ℝ$, so the hitting time $\tau$ will indeed lie in $𝕋$ whenever it is finite. The main cases are discrete-time when $𝕋={\mathbb{Z}}_{+}$ 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 set $\mathrm{T}$ is discrete, then the hitting time $\tau$ is a stopping time.

Proof.

For any $s\le t\in 𝕋$ then $X_s$ will be ${ℱ}_t/𝒜$-measurable, as it is adapted. So, by the fact that the $\sigma$-algebra ${ℱ}_t$ is closed under taking countable unions,

$$\{\tau \le t\}=\bigcup _{\begin{array}{c}s\in 𝕋,\\ s\le t\end{array}}{X}_s^{-1}(S)\in {ℱ}_t$$

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 continuous 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 $\tau$ of any closed subset $S\mathrm{\subseteq }E$ is a stopping time.

Proof.

We may suppose that $S$ is nonempty, and define the continuous function $d_S(x)\equiv inf\{d(x,y):y\in S\}$ on $E$. Then, $\tau$ is the first time at which $Y_t\equiv d_S(X_t)$ hits $0$. Letting $U$ be any countable and dense subset of $𝕋\cap [0,t]$ then the continuity of the sample paths of $Y$ gives,

$$\{\tau \le t\}=\{\underset{u\in U}{inf}{Y}_u=0\}.$$

As the infimum of a countable set of measurable functions is measurable, this shows that $\{\tau \le t\}$ is in ${ℱ}_t$. ∎

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 $\mathrm{(}{\mathrm{F}}_t\mathrm{)}$ is universally complete. Then, the hitting time $\tau$ of any closed subset $S\mathrm{\subseteq }E$ is a stopping time.

In particular, the hitting time of any closed set $S\subseteq ℝ$ 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 ${ℱ}_t$ is universally complete is necessary.

Progressively measurable processes

The début $D(A)$ of a set $A\subseteq 𝕋\times \mathrm{\Omega }$ is defined to be the hitting time of $\{1\}$ for the process $1_A$,

$$D(A)(\omega )=inf\{t\in 𝕋:(t,\omega )\in A\}.$$

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

Theorem (Début theorem).

Suppose that the filtration $\mathrm{(}{\mathrm{F}}_t\mathrm{)}$ is right-continuous and universally complete. Then, the début $D\mathit{}\mathrm{(}A\mathrm{)}$ of a progressively measurable $A\mathrm{\subseteq }\mathrm{T}\mathrm{\times }\mathrm{\Omega }$ is a stopping time.

Proofs of this typically rely upon properties of analytic sets, 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 $S\in 𝒜$. 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 $\mathrm{(}{\mathrm{F}}_t\mathrm{)}$ is right-continuous and universally complete, and that $X$ is a progressive process taking values in a measurable space $\mathrm{(}E\mathrm{,}\mathrm{A}\mathrm{)}$. Then, the hitting time $\tau$ of any set $S\mathrm{\in }\mathrm{A}$ 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