# Hitting time

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{{short description|Aspect of stochastic processes}}

thumb | right | alt=The hitting and stopping times of three samples of Brownian motion. | The Hitting times and stopping times of three samples of Brownian motion.
In the study of [stochastic processes](/source/stochastic_processes) in [mathematics](/source/mathematics), a '''hitting time''' (or '''first hit time''') is the first time at which a given process "hits" a given [subset](/source/subset) of the [state space](/source/state_space). '''Exit times''' and '''return times''' are also examples of hitting times.

==Definitions==

Let {{mvar|T}} be an ordered [index set](/source/index_set) such as the [natural number](/source/natural_number)s, {{tmath|\N,}} the non-negative [real number](/source/real_number)s, {{math|[0, +∞)}}, or a subset of these; elements {{tmath|t \in T}} can be thought of as "times". Given a [probability space](/source/probability_space) {{math|(Ω, Σ, Pr)}} and a [measurable state space](/source/measurable_space) {{mvar|S}}, let <math>X :\Omega \times T \to S</math> be a [stochastic process](/source/stochastic_process), and let {{mvar|A}} be a [measurable subset](/source/measurable_set) of the state space {{mvar|S}}. Then the '''first hit time''' <math>\tau_A : \Omega \to [0, +\infty]</math> is the [random variable](/source/random_variable) defined by

:<math>\tau_A (\omega) := \inf \{ t \in T \mid X_t (\omega) \in A \}.</math>

The '''first exit time''' (from {{mvar|A}}) is defined to be the first hit time for {{math|''S'' \ ''A''}}, the [complement](/source/complement_(set_theory)) of {{mvar|A}} in {{mvar|S}}. Confusingly, this is also often denoted by {{mvar|τ<sub>A</sub>}}.<ref>{{cite book 
| last = Øksendal
| first = Bernt K.
| authorlink = Bernt Øksendal
| title = Stochastic Differential Equations: An Introduction with Applications 
| edition = Sixth
| publisher=Springer
| location = Berlin 
| year = 2003 
| isbn = 978-3-540-04758-2
}}</ref>

The '''first return time''' is defined to be the first hit time for the [singleton](/source/singleton_(mathematics)) set {{math|{''X''<sub>0</sub>(''ω'')},}} which is usually a given deterministic element of the state space, such as the origin of the coordinate system.

==Examples==

* Any [stopping time](/source/stopping_time) is a hitting time for a properly chosen process and target set. This follows from the converse of the [Début theorem](/source/Hitting_time) (Fischer, 2013).
* Let {{mvar|B}} denote standard [Brownian motion](/source/Wiener_process) on the [real line](/source/real_line) {{tmath|\R}} starting at the origin. Then the hitting time {{mvar|τ<sub>A</sub>}} satisfies the measurability requirements to be a stopping time for every Borel measurable set {{tmath|A\subseteq\R.}}
* For {{mvar|B}} as above, let {{mvar|τ<sub>r</sub>}} ({{math|r > 0}}) denote the first exit time for the interval {{math|(&minus;''r'', ''r'')}}, i.e. the first hit time for <math>(-\infty,-r]\cup [r, +\infty).</math> Then the [expected value](/source/expected_value) and [variance](/source/variance) of {{mvar|τ<sub>r</sub>}} 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 {{mvar|B}} as above, the time of hitting a single point (different from the starting point 0) has the [Lévy distribution](/source/L%C3%A9vy_distribution).

* The [narrow escape problem](/source/narrow_escape_problem) considers the time it takes for a confined particle, undergoing Brownian motion, to escape through a small opening.

==Début theorem==

The hitting time of a set {{mvar|F}} is also known as the ''début'' of {{mvar|F}}. The Début theorem says that the hitting time of a measurable set {{mvar|F}}, for a [progressively measurable process](/source/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 process](/source/adapted_process)es.
The proof that the début is measurable is rather involved and involves properties of [analytic set](/source/analytic_set)s. The theorem requires the underlying probability space to be [complete](/source/complete_measure) or, at least, universally complete.

The ''converse of the Début theorem'' states that every [stopping time](/source/stopping_time) defined with respect to a [filtration](/source/Filtration_(probability_theory)) 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 {{math|{0} }} by this process is the considered stopping time. The proof is very simple.<ref name="Fischer (2013)">{{cite journal|last=Fischer|first=Tom|title=On simple representations of stopping times and stopping time sigma-algebras|journal=Statistics and Probability Letters|year=2013|volume=83|issue=1|pages=345–349|doi=10.1016/j.spl.2012.09.024|arxiv=1112.1603}}</ref>

==Markov chains==
If a [Markov chain](/source/Markov_chain) is irreducible and positive-recurrent, then the stationary distribution is unique and given by 

<math display=block>\begin{align}
\pi(i)=\frac{1}{\operatorname{E} \left[ \tau_i \right]}
\end{align}</math>

where {{mvar|τ<sub>i</sub>}} is the hitting time for a state {{mvar|i}}. <ref>{{cite book
 | last=Lawler
 | first=Gregory
 | title=Introduction to Stochastic Processes
 | edition=2nd
 | publisher=Chapman & Hall/CRC
 | year=2006
 | pages=24–25
 | isbn=978-1584886518
}}</ref>

This can be viewed as a special case of [Kac's lemma](/source/Kac's_lemma). 

==See also==
*[Cover time](/source/Cover_time), the time at which all states have been reached
*[Stopping time](/source/Stopping_time)

==References==
{{reflist}}

Category:Stochastic processes

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