Hitting time

In the study of stochastic processes in mathematics, a hitting time (or first hit time) is a particular instance of a stopping time, 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, [0, +∞), or a subset of these; elements "t" ∈ "T" can be thought of as "times". Given a probability space (Ω, Σ, Pr) and a measurable state space "S", let "X" : Ω × "T" → "S" be a stochastic process, and let "A" be a measurable subset of the state space "S". Then the first hit time "τ"_"A" : Ω → [0, +∞] is the random variable defined by

:$au\_\{A\}\; (omega)\; :=\; inf\; \{\; t\; in\; T\; |\; X\_\{t\}\; (omega)\; in\; A\; \}.$

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 "τ"_"A" (e.g. in Øksendal (2003)).

The first return time is defined to be the first hit time for the singleton set { "X"₀("ω") }, which is usually a given deterministic element of the state space, such as the origin of the coordinate system.

Example

Let "B" denote standard Brownian motion on the real line R starting at the origin. Then the hitting time "τ"_"A" satisfies the measurablility requirements to be a stopping time for every Borel measurable set "A" ⊆ R.

Let "τ"_"r", "r" > 0, denote the first exit time for the interval (−"r", "r"), i.e. the first hit time for (−∞, −"r"] ∪ ["r", +∞). Then the expected value and variance of "τ"_"r" satisfy

:$mathbb\{E\}\; left\; [\; au\_\{r\}\; ight]\; =\; r^\{2\},$:$mathrm\{Var\}\; left\; [\; au\_\{r\}\; ight]\; =\; (2/3)\; r^\{4\}.$

The time of hitting a single point (different from the starting point 0) has the Levy distribution.

Début theorem

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. The theorem requires the underlying probability space to be complete or, at least, universally complete.

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1