# hitting times are stopping times

Let ${({\mathcal{F}}_{t})}_{t\in \mathbb{T}}$ be a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space^{} $(\mathrm{\Omega},\mathcal{F})$. 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

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

$\tau (\omega )=inf\{t\in \mathbb{T}:{X}_{t}(\omega )\in S\}.$ |

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.

###### Proof.

For any $s\le t\in \mathbb{T}$ 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,

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

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 $S\mathrm{\subseteq}E$ 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 $\mathbb{T}\cap [0,t]$ 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 ${\mathcal{F}}_{t}$.
∎

## 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 $\mathrm{(}{\mathrm{F}}_{t}\mathrm{)}$ 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 $S\subseteq \mathbb{R}$ 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 $A\subseteq \mathbb{T}\times \mathrm{\Omega}$ is defined to be the hitting time of $\{1\}$ for the process ${1}_{A}$,

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

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 $(E,\mathcal{A})$ 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 $\mathrm{(}{\mathrm{F}}_{t}\mathrm{)}$ 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 |