In 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 numbers (or a subset of them) that is everywhere right-continuous and has left limits 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 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, and let . A function  is called a càdlàg function if, for every ,
 the left limit  exists; and
 the right limit  exists and equals .
That is,  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 functions are càdlàg functions. For instance the cumulative at point  correspond to the probability of being lower or equal than , namely . In other words, the semi-open interval of concern for a two-tailed distribution  is right-closed.
 The right derivative  of any convex function  defined on an open interval, is an increasing cadlag function.

Skorokhod space

The set of all càdlàg functions from  to  is often denoted by  (or simply ) and is called Skorokhod space after the 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"). For simplicity, take  and  — see Billingsley for a more general construction.

We must first define an analogue of the modulus of continuity, . For any , set
 
and, for , define the càdlàg modulus to be
 
where the infimum runs over all partitions , with . This definition makes sense for non-càdlàg  (just as the usual modulus of continuity makes sense for discontinuous functions) and it can be shown that  is càdlàg if and only if .

Now let  denote the set of all strictly increasing, continuous bijections from  to itself (these are "wiggles in time"). Let
 
denote the uniform norm on functions on . Define the Skorokhod metric  on  by
 
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 is indeed a metric. The topology  generated by  is called the Skorokhod topology on . 

An equivalent metric,
 
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  of continuous functions on  is a subspace of . The Skorokhod topology relativized to   coincides with the uniform topology there.

Completeness

It can be shown that, although  is not a complete space with respect to the Skorokhod metric , there is a topologically equivalent metric  with respect to which  is complete.

Separability

With respect to either  or ,  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  of probability measures on Skorokhod space  is tight if and only if both the following conditions are met:
 
and

Algebraic and topological structure
Under the Skorokhod topology and pointwise addition of functions,  is not a topological group, as can be seen by the following example:

Let  be a half-open interval and take  to be a sequence of characteristic functions.
Despite the fact that  in the Skorokhod topology, the sequence  does not converge to 0.

See also

References

Further reading 
 
 

Real analysis
Stochastic processes