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 is interchangeably either càdlàg or càglàd at each point of the domain.

Definition

Let be a metric space, and let. A function is called a càdlàg function if, for every,
That is, ƒ is right-continuous with left limits.

Examples

The set of all càdlàg functions from E to M 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 ƒ 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. Let
denote the uniform norm on functions on E. Define the Skorokhod metric σ on D 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 Σ generated by σ is called the Skorokhod topology on D.

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 with respect to the Skorokhod metric σ, there is a topologically equivalent metric σ0 with respect to which D is complete.

Separability

With respect to either σ or σ0, D 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 n=1,2,... of probability measures on Skorokhod space D 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, D is not a topological group, as can be seen by the following example:
Let be the unit 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.