Càdlàg

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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

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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  ,

  • the left limit   exists; and
  • the right limit   exists and equals  .

That is,   is right-continuous with left limits.

Examples

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  • 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

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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).   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

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Generalization of the uniform topology

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The space   of continuous functions on   is a subspace of  . The Skorokhod topology relativized to   coincides with the uniform topology there.

Completeness

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Although   is not a complete space with respect to the Skorokhod metric  , there is a topologically equivalent metric   with respect to which   is complete.[4]

Separability

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With respect to either   or  ,   is a separable space. Thus, Skorokhod space is a Polish space.

Tightness in Skorokhod space

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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

 

Algebraic and topological structure

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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.

See also

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References

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  1. ^ "Skorokhod space - Encyclopedia of Mathematics".
  2. ^ Billingsley, P. Convergence of Probability Measures. New York: Wiley.
  3. ^ Georgiou, T.T. and Smith, M.C. (2000). "Robustness of a relaxation oscillator". International Journal of Robust and Nonlinear Control. 10 (11–12): 1005–1024. doi:10.1002/1099-1239(200009/10)10:11/12<1005::AID-RNC536>3.0.CO;2-Q.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. ^ Billingsley, P. Convergence of Probability Measures. New York: Wiley.

Further reading

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