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 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 is interchangeably either càdlàg or càglàd at each point of the domain.
Let (M, d) be a metric space, and let E ⊆ R. A function ƒ: E → M is called a càdlàg function if, for every t ∈ E,
That is, ƒ is right-continuous with left limits.
The set of all càdlàg functions from E to M is often denoted by D(E; M) (or simply D) and is called Skorokhod space after the Soviet 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 E = [0, T] and M = Rn — see Billingsley for a more general construction.
We must first define an analogue of the modulus of continuity, ϖ′ƒ(δ). For any F ⊆ E, set
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