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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s (or a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of them) that is everywhere right-continuous and has left
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
s everywhere. Càdlàg functions are important in the study of
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
that admit (or even require) jumps, unlike
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, which has continuous sample paths. The collection of càdlàg functions on a given
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
is known as Skorokhod space. Two related terms are càglàd, standing for "continue à gauche, limite à droite", the left-right reversal of càdlàg, and càllàl for "continue à l'un, limite à l’autre" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd.


Definition

Let be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, and let . A function is called a càdlàg function if, for every , * the left limit exists; and * the right limit exists and equals ''f''(''t''). That is, ''f'' is right-continuous with left limits.


Examples

* All functions continuous on a subset of the real numbers are càdlàg functions on that subset. * As a consequence of their definition, all
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Eve ...
s are càdlàg functions. For instance the cumulative at point r correspond to the probability of being lower or equal than r, namely \mathbb \leq r/math>. In other words, the semi-open interval of concern for a two-tailed distribution (-\infty, r] is right-closed. * The right derivative f^\prime_+ of any
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poi ...
'' f'' defined on an open interval, is an increasing cadlag function.


Skorokhod space

The set of all càdlàg functions from ''E'' to ''M'' is often denoted by (or simply ''D'') and is called Skorokhod space after the Ukrainian mathematician
Anatoliy Skorokhod Anatoliy Volodymyrovych Skorokhod ( uk, Анато́лій Володи́мирович Скорохо́д; September 10, 1930January 3, 2011) was a Soviet and Ukrainian mathematician. Skorokhod is well-known for a comprehensive treatise on the ...
. Skorokhod space can be assigned a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that, intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
only allows us to "wiggle space a bit"). For simplicity, take and — see Billingsley for a more general construction. We must first define an analogue of the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f( ...
, . For any , set : w_ (F) := \sup_ , f(s) - f(t) , and, for , define the càdlàg modulus to be : \varpi'_ (\delta) := \inf_ \max_ w_ ( _, t_)), where the infimum runs over all partitions , , with . This definition makes sense for non-càdlàg ''ƒ'' (just as the usual modulus of continuity makes sense for discontinuous functions) and it can be shown that ''ƒ'' is càdlàg if and only if as . Now let Λ denote the set of all strictly increasing, continuous bijections from ''E'' to itself (these are "wiggles in time"). Let : \, f \, := \sup_ , f(t) , denote the uniform norm on functions on ''E''. Define the Skorokhod metric ''σ'' on ''D'' by : \sigma (f, g) := \inf_ \max \, where is the identity function. In terms of the "wiggle" intuition, measures the size of the "wiggle in time", and measures the size of the "wiggle in space". It can be shown that the Skorokhod
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
is indeed a metric. The topology Σ generated by ''σ'' is called the Skorokhod topology on ''D''. An equivalent metric, : d (f, g) := \inf_ (\, \lambda - I \, + \, f - g \circ \lambda \, ), was introduced independently and utilized in control theory for the analysis of switching systems.


Properties of Skorokhod space


Generalization of the uniform topology

The space ''C'' of continuous functions on ''E'' is a subspace of ''D''. The Skorokhod topology relativized to ''C'' coincides with the uniform topology there.


Completeness

It can be shown that, although ''D'' is not a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the Skorokhod metric ''σ'', there is a topologically equivalent metric ''σ''0 with respect to which ''D'' is complete.


Separability

With respect to either ''σ'' or ''σ''0, ''D'' is a
separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of th ...
. Thus, Skorokhod space is a
Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
.


Tightness in Skorokhod space

By an application of the Arzelà–Ascoli theorem, one can show that a sequence (''μn'')''n''=1,2,... of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s on Skorokhod space ''D'' is
tight Tight may refer to: Clothing * Skin-tight garment, a garment that is held to the skin by elastic tension * Tights, a type of leg coverings fabric extending from the waist to feet * Tightlacing, the practice of wearing a tightly-laced corset ...
if and only if both the following conditions are met: : \lim_ \limsup_ \mu_\big( \ \big) = 0, and : \lim_ \limsup_ \mu_\big( \ \big) = 0\text\varepsilon > 0.


Algebraic and topological structure

Under the Skorokhod topology and pointwise addition of functions, ''D'' is not a topological group, as can be seen by the following example: Let E=[0,2) be a half-open interval and take f_n = \chi_ \in D to be a sequence of characteristic functions. Despite the fact that f_n \rightarrow \chi_ in the Skorokhod topology, the sequence f_n - \chi_ does not converge to 0.


References


Further reading

* * {{DEFAULTSORT:Cadlag Real analysis Stochastic processes