Càdlàg


In mathematics, a càdlàg, RCLL, or corlol right, limit on function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit 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", 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,
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 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". 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 . is càdlàg if and only if.
Now let denote the set of all strictly increasing, continuous bijections from to itself. 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".
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

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.