In
mathematics, a
càdlàg (French "continue à droite, limitée 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
limit 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.
Definition
Let (M, d) be a
metric space, and let E
\subseteq \mathbb{R}. A function f : E \to M is called a
càdlàg function if, for every t \in E,
- the left limit f(t-) := \lim_{s
\uparrow t} f(s) exists; and
- the right limit f(t+) := \lim_{s
\downarrow t} f(s) exists and equals f(t).
That is, f is right-continuous with left limits.
Examples
Skorokhod space
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
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"). For
simplicity, take E = [0, T] and M = \mathbb{R}^{n} — see
Billingsley for a more general construction.
We must first define an analogue of the
modulus of continuity, \varpi'_{f}
(\delta). For any F \subseteq E, set
- w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |
and, for \delta > 0, define the
càdlàg modulus
to be
- \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
\Pi = \{ 0 = t_{0} t_{1} \dots t_{k} = T \}, k \in \mathbb{N}, with
\max_{i} (t_{i} - t_{i - 1}) \delta. This definition makes sense
for non-càdlàg f (just as the usual modulus of continuity makes
sense for discontinuous functions) and it can be shown that f is
càdlàg
if and only if \varpi'_{f}
(\delta) \to 0 as \delta \to 0.
Now let \Lambda denote the set of all
strictly increasing, continuous
bijections from E to itself (these are "wiggles in
time"). Let
- \| f \| := \sup_{t \in E} | f(t) |
denote the uniform norm on functions on E. Define the
Skorokhod metric \sigma on D by
- \sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda
- I \|, \| f - g \circ \lambda \| \},
where I : E \to E is the identity function. In terms of the
"wiggle" intuition, \| \lambda - I \| measures the size of the
"wiggle in time", and \| f - g \circ \lambda \| measures the size
of the "wiggle in space".
It can be shown that the Skorokhod
metric is, indeed a metric. The
topology \Sigma generated by \sigma is called the
Skorokhod
topology on D.
Properties of Skorokhod space
Generalization of the uniform topology
The space
C of continuous functions on
E is a
subspace of
D. The
Skorokhod topology relativized to
C coincides with the
uniform topology there.
Completeness
It can be shown that, although
D is not a
complete space with respect to the Skorokhod
metric
σ, there is a
topologically
equivalent metric σ_{0} with respect to which
D is complete.
Separability
With respect to either
σ or
σ_{0},
D is a
separable space.
Thus, Skorokhod space is a
Polish
space.
Tightness in Skorokhod space
By an application of the
Arzelà-Ascoli theorem, one can
show that a sequence (\mu_{n})_{n = 1}^{\infty} of
probability measures on Skorokhod space
D is
tight if and
only if both the following conditions are met:
- \lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in D |
\| f \| \geq a \} = 0,
and
- \lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in D |
\varpi'_{f} (\delta) \geq \varepsilon \} = 0\text{ for all
}\varepsilon > 0.
Algebraic and topological structure
Under the Skorokhod topology and pointwise addition of functions, D
is not a topological group.
References