hitting times are stopping times

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hitting times are stopping times

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

		$\displaystyle\tau\colon\Omega\rightarrow\mathbb{T}\cup\{\pm\infty\},$
		$\displaystyle\tau(\omega)=\inf\left\{t\in\mathbb{T}:X_{t}(\omega)\in S\right\}.$

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

Proof.

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

\left\{\tau\leq t\right\}=\bigcup_{{\substack{s\in\mathbb{T},\\ s\leq t}}}X_{s}^{{-1}}(S)\in\mathcal{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 $S\subseteq E$ is a stopping time.

Proof.

We may suppose that $S$ is nonempty, and define the continuous function $d_{S}(x)\equiv\inf\{d(x,y)\colon y\in S\}$ 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 $\mathbb{T}\cap[0,t]$ then the continuity of the sample paths of $Y$ gives,

\left\{\tau\leq t\right\}=\left\{\inf_{{u\in U}}Y_{u}=0\right\}.

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

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

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

Progressively measurable processes

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

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

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

Theorem (Début theorem).

Suppose that the filtration $(\mathcal{F}_{t})$ is right-continuous and universally complete. Then, the début $D(A)$ of a progressively measurable $A\subseteq\mathbb{T}\times\Omega$ 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 $(E,\mathcal{A})$ is said to be progressive if the set $X^{{-1}}(S)$ is progressively measurable for every $S\in\mathcal{A}$ . 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 $(\mathcal{F}_{t})$ is right-continuous and universally complete, and that $X$ is a progressive process taking values in a measurable space $(E,\mathcal{A})$ . Then, the hitting time $\tau$ of any set $S\in\mathcal{A}$ is a stopping time.

Defines:

hitting time,d\'ebut,debut,d\'ebut theorem,debut theorem

Keywords:

stopping time,adapted process,progressive process

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

stopping time

Mathematics Subject Classification

60G40 Stopping times; optimal stopping problems; gambling theory
60G05 Foundations of stochastic processes

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hitting times are stopping times

Primary tabs

hitting times are stopping times

Discrete-time processes

Theorem.

Proof.

Continuous processes

Theorem.

Proof.

Right-continuous processes

Theorem.

Progressively measurable processes

Theorem (Début theorem).

Theorem.

Mathematics Subject Classification

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hitting times are stopping times

Primary tabs

hitting times are stopping times

Discrete-time processes

Theorem.

Proof.

Continuous processes

Theorem.

Proof.

Right-continuous processes

Theorem.

Progressively measurable processes

Theorem (Début theorem).

Theorem.

Mathematics Subject Classification

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