View source for Càdlàg

{{Short description|Right continuous function with left limits}}
In [[mathematics]], a '''càdlàg''' ({{Langx|fr|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]]s (or a [[subset]] of them) that is everywhere [[right-continuous]] and has left [[Limit of a function|limit]]s everywhere. Càdlàg functions are important in the study of [[stochastic processes]] that admit (or even require) jumps, unlike [[Brownian motion]], which has continuous sample paths. The collection of càdlàg functions on a given [[domain of a function|domain]] is known as '''Skorokhod space'''.

Two related terms are '''càglàd''', standing for "{{Lang|fr|continue à gauche, limite à droite}}", the left-right reversal of càdlàg, and '''càllàl''' for "{{Lang|fr|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==
[[Image:Discrete probability distribution illustration.png|right|thumb|[[Cumulative distribution functions]] are examples of càdlàg functions.]]
[[File:Discrete probability distribution with a countable set of discontinuities.svg|right|thumb|Example of a cumulative distribution function with a countably infinite set of discontinuities]]

Let <math>(M, d)</math> be a [[metric space]], and let <math>E \subseteq \mathbb{R}</math>. A function <math>f:E \to M</math> is called a '''càdlàg function''' if, for every <math>t \in E</math>,
* the [[left limit]] <math>f(t-) := \lim_{s \to t^{-}}f(s)</math> exists; and
* the [[right limit]] <math>f(t+) := \lim_{s \to t^{+}}f(s)</math> exists and equals <math>f(t)</math>.
That is, <math>f</math> 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]]s are càdlàg functions. For instance the cumulative at point <math>r</math> correspond to the probability of being lower or equal than <math>r</math>, namely <math>\mathbb{P}[X\leq r]</math>. In other words, the semi-open [[interval (mathematics)|interval]] of concern for a two-tailed distribution <math>(-\infty, r]</math> is right-closed.
* The right derivative <math>f^\prime_+</math> of any [[convex function]] <math>f</math> defined on an open interval, is an increasing cadlag function.

==Skorokhod space==

The set of all càdlàg functions from <math>E</math> to <math>M</math> is often denoted by <math>\mathbb{D}(E:M)</math> (or simply <math>\mathbb{D}</math>) and is called '''Skorokhod space''' after the [[NASU Institute of Mathematics|Ukrainian mathematician]] [[Anatoliy Skorokhod]]. Skorokhod space can be assigned a [[topology]] that intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of [[uniform convergence]] only allows us to "wiggle space a bit").<ref>{{cite web | url=https://encyclopediaofmath.org/wiki/Skorokhod_space | title=Skorokhod space - Encyclopedia of Mathematics }}</ref> For simplicity, take <math>E = [0, T]</math> and <math>M = \mathbb{R}^n</math> — see Billingsley<ref>{{cite book |last1=Billingsley |first1=P. |title=Convergence of Probability Measures |publisher=Wiley |location=New York}}</ref> for a more general construction.

We must first define an analogue of the [[modulus of continuity]], <math>\varpi'_{f} (\delta)</math>. For any <math>F \subseteq E</math>, set
: <math>
    w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |
  </math>
and, for <math>\delta > 0</math>, define the '''càdlàg modulus''' to be
: <math>
    \varpi'_{f} (\delta) := \inf_{\Pi} \max_{1 \leq i \leq k} w_{f} ([t_{i - 1}, t_{i})),
  </math>
where the [[infimum]] runs over all partitions <math>\Pi = \{0 = t_0 < t_1 < \dots < t_k = T\},\; k \in E</math>, with <math>\min_i(t_i - t_{i+1}) > \delta</math>. This definition makes sense for non-càdlàg <math>f</math> (just as the usual modulus of continuity makes sense for discontinuous functions). <math>f</math> is càdlàg [[if and only if]] <math>\lim_{\delta \to 0} \varpi'_{f} (\delta) = 0</math>.

Now let <math>\Lambda</math> denote the set of all [[strictly increasing]], continuous [[bijection]]s from <math>E</math> to itself (these are "wiggles in time"). Let
: <math>
    \| f \| := \sup_{t \in E} | f(t) |
  </math>
denote the uniform norm on functions on <math>E</math>. Define the '''Skorokhod metric''' <math>\sigma</math> on <math>\mathbb{D}</math> by
: <math>
    \sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda - I \|, \| f - g \circ \lambda \| \},
  </math>
where <math>I: E \to E</math> is the identity function. In terms of the "wiggle" intuition, <math>\| \lambda - I \|</math> measures the size of the "wiggle in time", and <math>\| f - g \circ \lambda \|</math> measures the size of the "wiggle in space".

The Skorokhod [[Metric (mathematics)|metric]] is indeed a metric. The topology <math>\Sigma</math> generated by <math>\sigma</math> is called the '''Skorokhod topology''' on <math>\mathbb{D}</math>. 

An equivalent metric,
: <math>
    d (f, g) := \inf_{\lambda \in \Lambda}  (\| \lambda - I \|+ \| f - g \circ \lambda \|),
  </math>
was introduced independently and utilized in control theory for the analysis of switching systems.<ref name="georgiousmith">{{cite journal |author=Georgiou, T.T. and Smith, M.C.|title=Robustness of a relaxation oscillator |journal=International Journal of Robust and Nonlinear Control |volume=10  |issue=11–12 |pages=1005–1024 |year=2000|doi=10.1002/1099-1239(200009/10)10:11/12<1005::AID-RNC536>3.0.CO;2-Q }}</ref>

==Properties of Skorokhod space==

===Generalization of the uniform topology===

The space <math>C</math> of continuous functions on <math>E</math> is a [[Subspace topology|subspace]] of <math>\mathbb{D}</math>. The Skorokhod topology relativized to <math>C</math>  coincides with the uniform topology there.

===Completeness===

Although <math>\mathbb{D}</math> is not a [[complete space]] with respect to the Skorokhod metric <math>\sigma</math>, there is a [[Metric (mathematics)#Equivalence of metrics|topologically equivalent metric]] <math>\sigma_0</math> with respect to which <math>\mathbb{D}</math> is complete.<ref>{{cite book |last1=Billingsley |first1=P. |title=Convergence of Probability Measures |publisher=Wiley |location=New York}}</ref>

===Separability===

With respect to either <math>\sigma</math> or <math>\sigma_0</math>, <math>\mathbb{D}</math> is a [[separable space]]. Thus, Skorokhod space is a [[Polish space]].

===Tightness in Skorokhod space===
By an application of the [[Arzelà–Ascoli theorem]], one can show that a sequence <math>(\mu_n)_{n=1,2,\dots}</math> of [[probability measure]]s on Skorokhod space <math>\mathbb{D}</math> is [[tightness of measures|tight]] if and only if both the following conditions are met:
: <math>
    \lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in \mathbb{D} \;|\; \| f \| \geq a \} \big) = 0,
  </math>
and
: <math>
    \lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in \mathbb{D} \;|\; \varpi'_{f} (\delta) \geq \varepsilon \} \big) = 0\text{ for all }\varepsilon > 0.
  </math>

===Algebraic and topological structure===
Under the Skorokhod topology and pointwise addition of functions, <math>\mathbb{D}</math> is not a topological group, as can be seen by the following example:

Let <math> E=[0,2) </math> be a half-open interval and take <math>f_n = \chi_{[1-1/n,2)} \in \mathbb{D}</math> to be a sequence of characteristic functions.
Despite the fact that <math> f_n \rightarrow \chi_{[1,2)} </math> in the Skorokhod topology, the sequence <math> f_n - \chi_{[1,2)} </math> does not converge to 0.

==See also==

* {{annotated link|Classical Wiener space}}

==References==

{{reflist}}

==Further reading ==
* {{cite book | author=Billingsley, Patrick | title=Probability and Measure | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1995 | isbn=0-471-00710-2 |ref=none}}
* {{cite book | author=Billingsley, Patrick | title=Convergence of Probability Measures | url=https://archive.org/details/convergenceofpro0000bill | url-access=registration | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1999 | isbn=0-471-19745-9 |ref=none}}

{{DEFAULTSORT:Cadlag}}
[[Category:Real analysis]]
[[Category:Stochastic processes]]