Hitting time

The Hitting times and stopping times of three samples of Brownian motion.

In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times.

Definitions

Let Template:Mvar be an ordered index set such as the natural numbers, Template:Tmath the non-negative real numbers, Template:Math, or a subset of these; elements Template:Tmath can be thought of as "times". Given a probability space Template:Math and a measurable state space Template:Mvar, let $X:\mathrm{\Omega }\times T\to S$ be a stochastic process, and let Template:Mvar be a measurable subset of the state space Template:Mvar. Then the first hit time ${\tau }_A:\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ is the random variable defined by

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

The first exit time (from Template:Mvar) is defined to be the first hit time for Template:Math, the complement of Template:Mvar in Template:Mvar. Confusingly, this is also often denoted by Template:Mvar.

The first return time is defined to be the first hit time for the singleton set Template:Math which is usually a given deterministic element of the state space, such as the origin of the coordinate system.

Examples

• Any stopping time is a hitting time for a properly chosen process and target set. This follows from the converse of the Début theorem (Fischer, 2013).
• Let Template:Mvar denote standard Brownian motion on the real line Template:Tmath starting at the origin. Then the hitting time Template:Mvar satisfies the measurability requirements to be a stopping time for every Borel measurable set Template:Tmath
• For Template:Mvar as above, let Template:Mvar (Template:Math) denote the first exit time for the interval Template:Math, i.e. the first hit time for $(-\mathrm{\infty },-r]\cup [r,+\mathrm{\infty }).$ Then the expected value and variance of Template:Mvar satisfy

$$\begin{array}{cc}\mathrm{E}\left[{\tau }_r\right]& =r^2,\\ \mathrm{Var}\left[{\tau }_r\right]& =\frac{2}{3}{r}^4.\end{array}$$

• For Template:Mvar as above, the time of hitting a single point (different from the starting point 0) has the Lévy distribution.

Début theorem

The hitting time of a set Template:Mvar is also known as the début of Template:Mvar. The Début theorem says that the hitting time of a measurable set Template:Mvar, for a progressively measurable process with respect to a right continuous and complete filtration, is a stopping time. Progressively measurable processes include, in particular, all right and left-continuous adapted processes. The proof that the début is measurable is rather involved and involves properties of analytic sets. The theorem requires the underlying probability space to be complete or, at least, universally complete.

The converse of the Début theorem states that every stopping time defined with respect to a filtration over a real-valued time index can be represented by a hitting time. In particular, for essentially any such stopping time there exists an adapted, non-increasing process with càdlàg (RCLL) paths that takes the values 0 and 1 only, such that the hitting time of the set Template:Math by this process is the considered stopping time. The proof is very simple.

See also

• Stopping time