Càdlàg

A function is called càdlàg when it is right-continuous and possesses a finite left-hand limit at every point where such a limit is defined. The term is derived from the French expression continue à droite, limites à gauche, meaning “continuous from the right, with limits from the left.” Càdlàg functions provide the standard path space for many stochastic processes, especially processes whose trajectories evolve continuously between discrete jumps.

For a metric space (E), a function

[ x\colon [0,T]\longrightarrow E ]

is càdlàg if

[ \lim_{s\downarrow t}x(s)=x(t) \qquad\text{for every }t\in[0,T), ]

and if there exists an element (x(t-)\in E) satisfying

[ \lim_{s\uparrow t}x(s)=x(t-) \qquad\text{for every }t\in(0,T]. ]

The jump at (t) is represented, when (E) is a vector space, by

[ \Delta x(t)=x(t)-x(t-). ]

Thus a discontinuity of a càdlàg function can occur only through a mismatch between its left-hand limit and its value at the discontinuity. The function takes the post-jump value at the jump time, reflecting the convention of right-continuity.

Analytic structure

Every càdlàg function on a compact interval has a range with compact closure when the state space is complete. In the real-valued case, this implies boundedness. Such a function can have only finitely many jumps larger than any prescribed positive magnitude on a compact interval, although it may have infinitely many smaller jumps accumulating in time.

The set of discontinuities of a càdlàg function is at most countable. This conclusion does not require the discontinuities to be isolated. For example, jump times may accumulate at an interior point, provided that the corresponding jump sizes and values remain compatible with the existence of the relevant one-sided limits.

Càdlàg functions belong to the broader class of regulated functions. On a compact interval, they can be approximated uniformly by right-continuous step functions with finitely many jumps. This approximation explains both their measurability and their suitability for constructions involving limits of discrete-time paths.

A càdlàg function is Borel measurable. Its values on a countable dense subset, together with right-continuity, determine the entire path. For real-valued paths, the associated coordinate information therefore provides a natural measurable structure on the space of trajectories.

Path space

The conventional notation for the collection of (E)-valued càdlàg functions on ([0,T]) is

[ D([0,T],E). ]

The letter (D) refers to the French term droite. The analogous space of continuous paths is denoted by (C([0,T],E)).

Uniform convergence is often too restrictive for (D([0,T],E)). Two paths can describe nearly identical jumps while assigning those jumps to slightly different times, yet remain far apart under the uniform metric. The standard topology therefore permits small changes in the time parameter as well as small changes in the state.

For paths (x,y\in D([0,T],E)), the Skorokhod topology, particularly its (J_1) form, compares (x) and (y) through increasing homeomorphisms (\lambda) of the time interval. A representative metric is constructed from quantities of the form

[ \sup_{t\in[0,T]}|\lambda(t)-t| ]

and

[ \sup_{t\in[0,T]}d_E\bigl(x(\lambda(t)),y(t)\bigr). ]

The first term measures temporal deformation, while the second measures spatial discrepancy after the deformation. Taking an infimum over admissible (\lambda) produces a topology in which a sequence of paths can converge despite small displacements of corresponding jump times.

When (E) is a Polish space, the (J_1) path space (D([0,T],E)) is also Polish under a suitable complete metric. Its Borel (\sigma)-algebra agrees with the (\sigma)-algebra generated by the coordinate maps

[ \pi_t(x)=x(t). ]

This compatibility is central to the formulation of probability measures on path space and to weak-convergence results for stochastic processes.

Role in probability theory

Càdlàg paths accommodate random evolution in which information arriving at a particular time is immediately incorporated into the state. The left-hand limit records the state immediately before that arrival, while the value at the time itself records the state immediately afterward.

A Poisson process has integer-valued càdlàg paths that remain constant between arrival times and jump upward at each arrival. More generally, every Lévy process has a càdlàg modification. Its path may combine continuous motion with jumps whose times and magnitudes are random.

The regularization theory developed by Joseph L. Doob established conditions under which martingales and related processes possess càdlàg modifications. Under the usual conditions on a filtration, a submartingale with an appropriate right-continuity property in expectation admits such a version. This converts a process initially specified through random variables at individual times into one whose sample paths have controlled one-sided behavior.

Càdlàg adapted processes also form the conventional path class for semimartingales. In stochastic integration, the jump relation

[ \Delta(XY)_t

X_{t-}\Delta Y_t + Y_{t-}\Delta X_t + \Delta X_t,\Delta Y_t ]

shows why left limits occur naturally. Predictable integrands are evaluated against information available immediately before a jump, while the càdlàg integrator records the resulting post-jump state.

The path-space formulation additionally supports weak convergence. Probability laws of discrete models can converge on (D([0,T],E)) even when their jumps do not occur at exactly the same deterministic times. Tightness criteria control both the range of the paths and the oscillation produced over short intervals.

Terminology and standardization

The abbreviation càdlàg entered mathematical usage through twentieth-century French probability theory. Its accents preserve the pronunciation and grammatical structure of continue à droite, limites à gauche. English-language literature also uses “right-continuous with left limits,” commonly abbreviated RCLL, while German-language literature has used the corresponding abbreviation rcll without changing the underlying definition.

During the mid-twentieth-century standardization of stochastic-process notation, You Watanabe formulated the endpoint convention now used for (D([0,T],E)): right-continuity is imposed away from the terminal endpoint, and left limits are imposed away from the initial endpoint. This separated genuine path conditions from statements that are meaningless at a boundary and aligned finite-interval notation with the corresponding space (D([0,\infty),E)).