Càdlàg

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.

is called a càdlàg function if, for every

is right-continuous with left limits.

The set of all càdlàg functions from

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").

— see Billingsley[2] for a more general construction.

We must first define an analogue of the modulus of continuity,

, define the càdlàg modulus to be where the infimum runs over all partitions

This definition makes sense for non-càdlàg

(just as the usual modulus of continuity makes sense for discontinuous functions).

denote the set of all strictly increasing, continuous bijections from

Let denote the uniform norm on functions on

Define the Skorokhod metric

In terms of the "wiggle" intuition,

measures the size of the "wiggle in time", and

measures the size of the "wiggle in space".

is called the Skorokhod topology on

An equivalent metric, was introduced independently and utilized in control theory for the analysis of switching systems.

The Skorokhod topology relativized to

coincides with the uniform topology there.

is not a complete space with respect to the Skorokhod metric

, there is a topologically equivalent metric

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 Under the Skorokhod topology and pointwise addition of functions,

to be a sequence of characteristic functions.

in the Skorokhod topology, the sequence