Hitting time
Encyclopedia
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.
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 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 τA : Ω → [0, +∞] 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 (e.g. in Øksendal (2003)).
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.
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 ⊆ R.
Let
(
) 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
satisfy
The time of hitting a single point (different from the starting point 0) has the Lévy distribution.
, is a stopping time. Progressively measurable processes include, in particular, all right and left-continuous adapted process
es.
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
or, at least, universally complete.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a hitting time (or first hit time) is a particular instance of a stopping time
Stopping rule
In probability theory, in particular in the study of stochastic processes, a stopping time is a specific type of “random 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 setIndex set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
such as the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, N, the non-negative real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, [0, +∞), or a subset of these; elements t ∈ T can be thought of as "times". Given a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
(Ω, Σ, Pr) and a measurable state space S, let X : Ω × T → S be a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
, and let A be a measurable subset of the state space S. Then the first hit time τA : Ω → [0, +∞] is the random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
defined by
The first exit time (from A) is defined to be the first hit time for S \ A, the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of A in S. Confusingly, this is also often denoted by τA (e.g. in Øksendal (2003)).
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.
Example
Let B denote standard Brownian motionWiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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 ⊆ R.
Let
() denote the first exit time for the interval (−r, r), i.e. the first hit time for (−∞, −r] ∪ [r, +∞). Then the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of
satisfyThe time of hitting a single point (different from the starting point 0) has the Lévy 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 processProgressively measurable process
In mathematics, progressive measurability is a property of stochastic processes. A progressively measurable process is one for which events defined in terms of values of the process across a range of times can be assigned probabilities . Being progressively measurable is a strictly stronger...
, is a stopping time. Progressively measurable processes include, in particular, all right and left-continuous adapted process
Adapted process
In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...
es.
The proof that the début is measurable is rather involved and involves properties of analytic set
Analytic set
In descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .- Definition :There are several equivalent definitions of analytic set...
s. The theorem requires the underlying probability space to be complete
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
or, at least, universally complete.