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Càdlàg

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[edit]

Let ${\displaystyle (M,d)}$ be a metric space, and let ${\displaystyle E\subseteq \mathbb {R} }$. A function ${\displaystyle f:E\to M}$ is called a càdlàg function if, for every ${\displaystyle t\in E}$,

• the left limit ${\displaystyle f(t-):=\lim _{s\to t^{-}}f($s)} exists; and
• the right limit ${\displaystyle f(t+):=\lim _{s\to t^{+}}f($s)} exists and equals ${\displaystyle f($t)} .

That is, ${\displaystyle f}$ is right-continuous with left limits.

Examples[edit]

• 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 ${\displaystyle r}$ correspond to the probability of being lower or equal than ${\displaystyle r}$, namely ${\displaystyle \mathbb {P} [X\leq r]}$. In other words, the semi-open interval of concern for a two-tailed distribution ${\displaystyle (-\infty ,r]}$ is right-closed.
• The right derivative ${\displaystyle f_{+}^{\prime }}$ of any convex function ${\displaystyle f}$ defined on an open interval, is an increasing cadlag function.

Skorokhod space[edit]

The set of all càdlàg functions from ${\displaystyle E}$to${\displaystyle M}$ is often denoted by ${\displaystyle \mathbb {D} (E:M)}$ (or simply ${\displaystyle \mathbb {D} }$) 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 ${\displaystyle E=[0,T]}$ and ${\displaystyle M=\mathbb {R} ^{n}}$ — see Billingsley^[2] for a more general construction.

We must first define an analogue of the modulus of continuity, ${\displaystyle \varpi '_{f}(\delta )}$. For any ${\displaystyle F\subseteq E}$, set

${\displaystyle w_{f}($F):=\sup _{s,t\in F}|f(s)-f(t)|}

and, for ${\displaystyle \delta >0}$, define the càdlàg modulus to be

${\displaystyle \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),}$

where the infimum runs over all partitions ${\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\},\;k\in E}$, with ${\displaystyle \min _{i}(t_{i}-t_{i+1})>\delta }$. This definition makes sense for non-càdlàg ${\displaystyle f}$ (just as the usual modulus of continuity makes sense for discontinuous functions). ${\displaystyle f}$ is càdlàg if and only if ${\displaystyle \lim _{\delta \to 0}\varpi '_{f}(\delta )=0}$.

Now let ${\displaystyle \Lambda }$ denote the set of all strictly increasing, continuous bijections from ${\displaystyle E}$ to itself (these are "wiggles in time"). Let

${\displaystyle \|f\|:=\sup _{t\in E}|f($t)|}

denote the uniform norm on functions on ${\displaystyle E}$. Define the Skorokhod metric ${\displaystyle \sigma }$on${\displaystyle \mathbb {D} }$by

${\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},}$

where ${\displaystyle I:E\to E}$ is the identity function. In terms of the "wiggle" intuition, ${\displaystyle \|\lambda -I\|}$ measures the size of the "wiggle in time", and ${\displaystyle \|f-g\circ \lambda \|}$ measures the size of the "wiggle in space".

The Skorokhod metric is indeed a metric. The topology ${\displaystyle \Sigma }$ generated by ${\displaystyle \sigma }$ is called the Skorokhod topologyon${\displaystyle \mathbb {D} }$.

An equivalent metric,

${\displaystyle d(f,g):=\inf _{\lambda \in \Lambda }(\|\lambda -I\|+\|f-g\circ \lambda \|),}$

was introduced independently and utilized in control theory for the analysis of switching systems.^[3]

Properties of Skorokhod space[edit]

Generalization of the uniform topology[edit]

The space ${\displaystyle C}$ of continuous functions on ${\displaystyle E}$ is a subspaceof${\displaystyle \mathbb {D} }$. The Skorokhod topology relativized to ${\displaystyle C}$ coincides with the uniform topology there.

Completeness[edit]

Although ${\displaystyle \mathbb {D} }$ is not a complete space with respect to the Skorokhod metric ${\displaystyle \sigma }$, there is a topologically equivalent metric ${\displaystyle \sigma _{0}}$ with respect to which ${\displaystyle \mathbb {D} }$ is complete.^[4]

Separability[edit]

With respect to either ${\displaystyle \sigma }$or${\displaystyle \sigma _{0}}$, ${\displaystyle \mathbb {D} }$ is a separable space. Thus, Skorokhod space is a Polish space.

Tightness in Skorokhod space[edit]

By an application of the Arzelà–Ascoli theorem, one can show that a sequence ${\displaystyle (\mu _{n})_{n=1,2,\dots }}$ofprobability measures on Skorokhod space ${\displaystyle \mathbb {D} }$istight if and only if both the following conditions are met:

${\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\|f\|\geq a\}{\big )}=0,}$

and

${\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\varpi '_{f}(\delta )\geq \varepsilon \}{\big )}=0{\text{ for all }}\varepsilon >0.}$

Algebraic and topological structure[edit]

Under the Skorokhod topology and pointwise addition of functions, ${\displaystyle \mathbb {D} }$ is not a topological group, as can be seen by the following example:

Let ${\displaystyle E=[0,2)}$ be a half-open interval and take ${\displaystyle f_{n}=\chi _{[1-1/n,2)}\in \mathbb {D} }$ to be a sequence of characteristic functions. Despite the fact that ${\displaystyle f_{n}\rightarrow \chi _{[1,2)}}$ in the Skorokhod topology, the sequence ${\displaystyle f_{n}-\chi _{[1,2)}}$ does not converge to 0.

See also[edit]

• Classical Wiener space

References[edit]

Further reading[edit]

• Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
• Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.