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| 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.
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| ==Definitions==
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| Let ''T'' be an ordered [[index set]] such as the [[natural number]]s, '''N''', the non-negative [[real number]]s, [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 space|measurable state space]] ''S'', let ''X'' : Ω × ''T'' → ''S'' be a [[stochastic process]], and let ''A'' be a [[measurable set|measurable subset]] of the state space ''S''. Then the '''first hit time''' ''τ''<sub>''A''</sub> : Ω → [0, +∞] is the [[random variable]] defined by
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| :<math>\tau_{A} (\omega) := \inf \{ t \in T | X_{t} (\omega) \in A \}.</math>
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| The '''first exit time''' (from ''A'') is defined to be the first hit time for ''S'' \ ''A'', the [[complement (set theory)|complement]] of ''A'' in ''S''. Confusingly, this is also often denoted by ''τ''<sub>''A''</sub> (e.g. in Øksendal (2003)).
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| The '''first return time''' is defined to be the first hit time for the [[singleton (mathematics)|singleton]] set { ''X''<sub>0</sub>(''ω'') }, which is usually a given deterministic element of the state space, such as the origin of the coordinate system.
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| ==Examples==
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| * Any [[stopping time]] is a hitting time for a properly chosen process and target set. This follows from the converse of the [[Hitting time#Début theorem|Début theorem]] (Fischer, 2013).
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| * Let ''B'' denote standard [[Wiener process|Brownian motion]] on the [[real line]] '''R''' starting at the origin. Then the hitting time ''τ''<sub>''A''</sub> satisfies the measurability requirements to be a stopping time for every Borel measurable set ''A'' ⊆ '''R'''.
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| * For ''B'' as above, let <math>\tau_r</math> (<math>r>0</math>) 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 <math>\tau_r</math> satisfy
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| :<math>\mathbb{E} \left[ \tau_{r} \right] = r^{2},</math> | |
| :<math>\mathrm{Var} \left[ \tau_{r} \right] = (2/3) r^{4}.</math>
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| * For ''B'' as above, the time of hitting a single point (different from the starting point 0) has the [[Lévy distribution]].
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| ==Début theorem==
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| 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 process]]es.
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| The proof that the début is measurable is rather involved and involves properties of [[analytic set]]s. The theorem requires the underlying probability space to be [[complete measure|complete]] or, at least, universally complete.
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| The ''converse of the Début theorem'' states that every [[stopping time]] defined with respect to a [[Filtration (mathematics)|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 <math> \{ 0 \}</math> by this process is the considered stopping time. The proof is very simple (Fischer, 2013).
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| ==See also==
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| *[[Stopping time]]
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| ==References==
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| * {{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|url=http://dx.doi.org/10.1016/j.spl.2012.09.024}}
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| * {{cite book
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| | last = Øksendal
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| | first = Bernt K.
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| | authorlink = Bernt Øksendal
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| | title = Stochastic Differential Equations: An Introduction with Applications
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| | edition = Sixth edition
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| | publisher=Springer
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| | location = Berlin
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| | year = 2003
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| | isbn = 3-540-04758-1
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| }}
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| [[Category:Stochastic processes]]
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