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.
Definition
Cumulative distribution functions are examples of càdlàg functions.Example of a cumulative distribution function with a countably infinite set of discontinuities
Let be a
metric space, and let . A function is called a càdlàg function if, for every ,
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 (or simply ) and is called Skorokhod space after the
Ukrainian mathematicianAnatoliy 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.
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). 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".
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]
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.
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.