In the study of stochastic processes in mathematics, a hitting 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 T be an ordered index set such as the natural numbers, N, the non-negative real numbers, is the random variable defined by The first exit time is defined to be the first hit time for S \ A, the complement of A in S. Confusingly, this is also often denoted by τA. The first return time is defined to be the first hit time for the singleton set, which is usually a given deterministic element of the state space, such as the origin of the coordinate system.
Let B denote standard Brownian motion on the real lineR starting at the origin. Then the hitting time τA satisfies the measurability requirements to be a stopping time for every Borel measurable setA ⊆ R.
For B as above, let denote the first exit time for the interval, i.e. the first hit time for. Then the expected value and variance of satisfy
For B as above, the time of hitting a single point has the Lévy distribution.
The hitting time of a set F is also known as the début of F. The Début theorem says that the hitting time of a measurable set F, for a progressively measurable process, is a stopping time. Progressively measurable processes include, in particular, all right and left-continuous adaptedprocesses. 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 paths that takes the values 0 and 1 only, such that the hitting time of the set by this process is the considered stopping time. The proof is very simple.
