Hitting time

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. Let T be an ordered index set such as the natural numbers, \N, the non-negative real numbers, [0, +∞), or a subset of these; elements t \in T can be thought of as "times". Given a probability space (Ω, Σ, Pr) and a measurable state space S, let be a stochastic process, and let A be a measurable subset of the state space S. Then the first hit time is the random variable defined by The first exit time (from A) 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 {X0(ω)}, which is usually a given deterministic element of the state space, such as the origin of the coordinate system. 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 B denote standard Brownian motion on the real line \R starting at the origin. Then the hitting time τA satisfies the measurability requirements to be a stopping time for every Borel measurable set A\subseteq\R. For B as above, let τr (r > 0) denote the first exit time for the interval (−r, r), i.e. the first hit time for Then the expected value and variance of τr satisfy For B as above, the time of hitting a single point (different from the starting point 0) 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 adapted processes. The proof that the début is measurable is rather involved and involves properties of analytic sets.