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

Discrete-time processes

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

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, \begin{equation*} \left\{\tau\le t\right\}=\bigcup_{\substack{s\in\mathbb{T},\\ s\le t}}X_s^{-1}(S)\in\mathcal{F}_t \end{equation*}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.

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, \begin{equation*} \left\{\tau\le t\right\}=\left\{\inf_{u\in U}Y_u=0\right\}. \end{equation*}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.

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 $1_A$ , \begin{equation*} D(A)(\omega)=\inf\left\{ t\in\mathbb{T}:(t,\omega)\in A\right\}. \end{equation*}An important result for continuous-time stochastic processes is the début theorem.

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.