Английская Википедия: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.
Definitions
Let Шаблон:Mvar be an ordered index set such as the natural numbers, Шаблон:Tmath the non-negative real numbers, Шаблон:Math, or a subset of these; elements Шаблон:Tmath can be thought of as "times". Given a probability space Шаблон:Math and a measurable state space Шаблон:Mvar, let <math>X :\Omega \times T \to S</math> be a stochastic process, and let Шаблон:Mvar be a measurable subset of the state space Шаблон:Mvar. Then the first hit time <math>\tau_A : \Omega \to [0, +\infty]</math> is the random variable defined by
- <math>\tau_A (\omega) := \inf \{ t \in T \mid X_t (\omega) \in A \}.</math>
The first exit time (from Шаблон:Mvar) is defined to be the first hit time for Шаблон:Math, the complement of Шаблон:Mvar in Шаблон:Mvar. Confusingly, this is also often denoted by Шаблон:Mvar.[1]
The first return time is defined to be the first hit time for the singleton set Шаблон: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 Шаблон:Mvar denote standard Brownian motion on the real line Шаблон:Tmath starting at the origin. Then the hitting time Шаблон:Mvar satisfies the measurability requirements to be a stopping time for every Borel measurable set Шаблон:Tmath
- For Шаблон:Mvar as above, let Шаблон:Mvar (Шаблон:Math) denote the first exit time for the interval Шаблон:Math, i.e. the first hit time for <math>(-\infty,-r]\cup [r, +\infty).</math> Then the expected value and variance of Шаблон:Mvar satisfy
<math display=block>\begin{align} \operatorname{E} \left[ \tau_r \right] &= r^2, \\ \operatorname{Var} \left[ \tau_r \right] &= \tfrac{2}{3} r^4. \end{align}</math>
- For Шаблон: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 Шаблон:Mvar is also known as the début of Шаблон:Mvar. The Début theorem says that the hitting time of a measurable set Шаблон:Mvar, 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.
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 Шаблон:Math by this process is the considered stopping time. The proof is very simple.[2]
See also
References
