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.
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").[1] For simplicity, take and — see Billingsley[2] 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.[3]
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.