# Càdlàg function

A **càdlàg function** is a function defined on an interval of the real line that is right-continuous at every point and has a left limit at every point, where the left limit need not equal the function's value. The name is a French acronym for *continue à droite, limite à gauche*, "continuous on the right, limit on the left"<sup>[1](https://web.stanford.edu/~hammond/Probab20B.pdf)</sup>. The condition is the standard regularity assumed for the sample paths of continuous-time stochastic processes that are allowed to jump, including Lévy processes, Poisson processes, and semimartingales<sup>[2](https://riskhub.org/maths-for-quants/course-content/stochastic-calculus-advanced-applications/advanced-stochastic-processes/levy-processes-and-applications-in-finance-2166)</sup><sup> • </sup><sup>[3](https://www.cambridge.org/core/books/abs/stochastic-processes/poisson-process/272BE8BD2D8234ACE499E8EEFFBB2A59)</sup>.

| Key fact | Detail |
|---|---|
| Definition | Right-continuous at every point, with a left limit existing at every point; the two conditions are independent, and a jump is a point where the left limit differs from the value<sup>[1](https://web.stanford.edu/~hammond/Probab20B.pdf)</sup><sup> • </sup><sup>[4](https://androma.org/page/C%C3%A0dl%C3%A0g%20Function)</sup> |
| Etymology | French acronym *continue à droite, limite à gauche*; equivalent English terms are RCLL (right-continuous with left limits), R-process, and right process<sup>[1](https://web.stanford.edu/~hammond/Probab20B.pdf)</sup><sup> • </sup><sup>[5](https://almostsuremath.com/2009/12/18/cadlag-modifications/)</sup> |
| Canonical example | Every cumulative distribution function is càdlàg: right-continuity holds by definition of F(x) as a probability of the event {X ≤ x}, and left limits exist by monotonicity<sup>[1](https://web.stanford.edu/~hammond/Probab20B.pdf)</sup> |
| Jump notation | For a càdlàg path X, the left limit is written X(t−) and the jump size is ΔX(t) = X(t) − X(t−)<sup>[6](https://math.berkeley.edu/~ktawri/Notes.pdf)</sup> |
| Path space | The set of càdlàg paths is the Skorokhod space D, introduced by A. V. Skorokhod in his 1956 paper "Limit theorems for stochastic processes"<sup>[7](https://www.mathnet.ru/php/archive.phtml?jrnid=tvp&option_lang=eng&paperid=5003&wshow=paper)</sup><sup> • </sup><sup>[8](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)</sup> |
| Modification theorems | A martingale with respect to a right-continuous filtration always has a càdlàg modification; stochastic continuity of a Lévy process already guarantees a càdlàg modification exists<sup>[9](https://link.springer.com/article/10.1007/s00591-023-00353-2)</sup><sup> • </sup><sup>[10](https://jovanaeducation.com/glossary/cadlag-sample-paths)</sup> |
| Mirror term | Càglàd, left-continuous with right limits, is the corresponding condition used for integrands in stochastic calculus<sup>[11](https://people.math.wisc.edu/~tseppalainen/courses/735/notes.pdf)</sup><sup> • </sup><sup>[12](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Cadlag.html)</sup> |

## Definition

Let T > 0 and let E be a metric space with metric d. A function a: [0, T] → E is càdlàg on [0, T] if two conditions hold<sup>[4](https://androma.org/page/C%C3%A0dl%C3%A0g%20Function)</sup>:

1. For every t ∈ [0, T), \( \lim_{s \downarrow t} d(a(s), a(t)) = 0 \): the function is continuous when approached from the right.
2. For every t ∈ (0, T], there exists a point a(t−) ∈ E with \( \lim_{s \uparrow t} d(a(s), a(t-)) = 0 \): the left limit exists.

When E = ℝ this says exactly that a is continuous from the right and has finite left limits at every positive time<sup>[4](https://androma.org/page/C%C3%A0dl%C3%A0g%20Function)</sup>. The two requirements are separate: a function can be right-continuous without its left limits existing, and a function with left limits everywhere can fail to be right-continuous. A jump at t is the quantity ΔX(t) = X(t) − X(t−), the gap between the value and the left limit<sup>[6](https://math.berkeley.edu/~ktawri/Notes.pdf)</sup>.

The definition extends to functions on [0, ∞), and the space of such functions with values in a set E is written D(ℝ₊, E), in contrast to C(ℝ₊, E) for continuous functions<sup>[8](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)</sup>.

## Why right-continuity

The asymmetry in the definition is a convention with a purpose. Right-continuity selects the post-jump value as the value of the path at the jump time, which is the convention that adapted stochastic processes need when values are observed through a filtration, the growing family of σ-algebras that records information available up to each time<sup>[4](https://androma.org/page/C%C3%A0dl%C3%A0g%20Function)</sup>. At a jump the path snaps to its new value immediately, but the left limit still records where it came from<sup>[2](https://riskhub.org/maths-for-quants/course-content/stochastic-calculus-advanced-applications/advanced-stochastic-processes/levy-processes-and-applications-in-finance-2166)</sup>.

The cumulative distribution function illustrates the same asymmetry from the other side. Because F(x) is defined as the probability of the event {X ≤ x}, a point mass at x sits inside F(x) but not inside the left limit, so F is right-continuous but generally not left-continuous; the probability of the exact value x is the jump size F(x) − lim_{h↑0} F(x + h)<sup>[1](https://web.stanford.edu/~hammond/Probab20B.pdf)</sup>.

## Where càdlàg paths appear

**Jump processes.** The Poisson process, the prototype of a pure jump process and the building block for Lévy processes, is defined with paths that are right-continuous with left limits<sup>[3](https://www.cambridge.org/core/books/abs/stochastic-processes/poisson-process/272BE8BD2D8234ACE499E8EEFFBB2A59)</sup>. A [Lévy process](https://www.edgepedia.org/levy-process), a process with stationary independent increments starting at zero and continuous in probability, has càdlàg sample paths: jumps occur at random times, never at a fixed pre-known clock tick, and the path is right-continuous with left limits<sup>[2](https://riskhub.org/maths-for-quants/course-content/stochastic-calculus-advanced-applications/advanced-stochastic-processes/levy-processes-and-applications-in-finance-2166)</sup>.

**Martingales and semimartingales.** A sub- or supermartingale with respect to a right-continuous filtration has a càdlàg modification whenever t ↦ E[M_t] is right-continuous; in particular, a martingale with respect to a right-continuous filtration always has a càdlàg modification<sup>[9](https://link.springer.com/article/10.1007/s00591-023-00353-2)</sup>. The definition of a semimartingale, a process decomposable as X = M + A with M an L²-martingale and A a finite-variation process, is stated for càdlàg paths, and the stochastic integral is developed for predictable integrands against càdlàg semimartingale integrators<sup>[6](https://math.berkeley.edu/~ktawri/Notes.pdf)</sup><sup> • </sup><sup>[13](https://people.math.wisc.edu/~tseppalainen/sa-book/etusivu.html)</sup>. In the stochastic calculus for jump processes the integrand must be left-continuous while the integrator is right-continuous for the integral to retain the martingale property<sup>[14](https://russell-davidson.research.mcgill.ca/e765/index.html)</sup>, which is where the càglàd condition on integrands enters<sup>[11](https://people.math.wisc.edu/~tseppalainen/courses/735/notes.pdf)</sup>.

**Measurability.** The càdlàg assumption also does measure-theoretic work. A jointly measurable map (ω, t) ↦ X_t(ω) need not be measurable for the product σ-algebra in continuous time, but a right-continuous path with left limits is the pointwise limit of the countably-valued approximations X_{2⁻ⁿ⌈2ⁿt⌉}, and is therefore measurable; continuous and càdlàg functions are moreover uniquely determined by their values on a countable dense set<sup>[8](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)</sup>.

## Skorokhod space and its topologies

The set of càdlàg paths, sometimes called the Skorokhod space and denoted D[0, 1] or D(ℝ₊, E), is the canonical state space for jump processes<sup>[8](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)</sup><sup> • </sup><sup>[15](https://www.sciencedirect.com/science/article/abs/pii/S0888613X22000755)</sup>. Weak convergence of processes with jumps cannot be handled by uniform convergence, because a sequence of jump paths converging to a jump limit cannot converge uniformly. A. V. Skorokhod introduced four metrics on D[0, 1] in 1956, called J1, M1, J2, and M2<sup>[16](https://www.math.sinica.edu.tw/bulletins/20131/2013106.pdf)</sup><sup> • </sup><sup>[7](https://www.mathnet.ru/php/archive.phtml?jrnid=tvp&option_lang=eng&paperid=5003&wshow=paper)</sup>. The standard J1 metric allows small reparametrizations of time,

\[ d_{J_1}(f, g) = \inf_{\lambda} \left\{ \| f \circ \lambda - g \|_{\infty} + \| \lambda - \mathrm{id} \|_{\infty} \right\}, \]

where the infimum runs over increasing bijections λ of [0, 1]<sup>[9](https://link.springer.com/article/10.1007/s00591-023-00353-2)</sup>. Endowed with this metric, D[0, 1] is a complete separable metric space, so [Prokhorov's theorem](https://www.edgepedia.org/prokhorovs-theorem) applies and limit theorems such as [Donsker's theorem](https://www.edgepedia.org/donskers-theorem) can be proved on path space<sup>[17](https://encyclopediaofmath.org/wiki/Skorokhod_topology)</sup><sup> • </sup><sup>[9](https://link.springer.com/article/10.1007/s00591-023-00353-2)</sup>. The weaker M1 topology is used when the candidate limit is discontinuous in ways J1 cannot reach, for example monotone processes with vanishing jumps converging to a process with jumps<sup>[18](https://www.cmap.polytechnique.fr/preprint/repository/771.pdf)</sup>.

## Modifications and existence

Many processes are first specified only through finite-dimensional distributions, for which the Daniell-[Kolmogorov extension theorem](https://www.edgepedia.org/kolmogorov-extension-theorem) constructs a probability measure whose product σ-algebra is too poor to express path properties. The standard remedy is to modify the process, changing the sample paths on a null set so that they become càdlàg<sup>[19](https://arxiv.org/abs/2301.07992)</sup>. Conditions for the existence of measurable, càdlàg, and continuous modifications are a standard topic in graduate probability<sup>[20](https://www.stat.cmu.edu/~cshalizi/754/2006/)</sup>.

The guarantee is not automatic. Strong Feller property and joint space-time continuity of a semigroup of Markov kernels on a [Polish space](https://www.edgepedia.org/polish-space) are not enough to ensure an associated càdlàg Markov process on the same space; an elementary counterexample is the Brownian semigroup on ℝ restricted to ℝ \ {0}, for which no associated càdlàg Markov process exists<sup>[21](https://arxiv.org/abs/2211.14789)</sup>.

## Terminology and formalization

Alternative names for the same property include rcll, [R-process](https://www.edgepedia.org/r-process), and right process; Rogers and Williams' *Diffusions, Markov Processes, and Martingales* calls such paths Skorokhod processes<sup>[5](https://almostsuremath.com/2009/12/18/cadlag-modifications/)</sup><sup> • </sup><sup>[22](https://assets.cambridge.org/97805217/75939/excerpt/9780521775939_excerpt.pdf)</sup>. The mirror condition, left-continuous with right limits, is written càglàd, from *continu à gauche, limite à droite*<sup>[11](https://people.math.wisc.edu/~tseppalainen/courses/735/notes.pdf)</sup>. The definition has also been formalized in the Lean mathematical library Mathlib, which defines right-continuity as f x → f a when x → a with x > a, together with the càglàd dual<sup>[12](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Cadlag.html)</sup>.

## References

1. [Peter J. Hammond, Lecture Notes 9: Probability, Part B: Probability as Measure, Stanford University](https://web.stanford.edu/~hammond/Probab20B.pdf)
2. [Lévy Processes and Applications in Finance, Risk Hub, Stochastic Calculus: Advanced Applications](https://riskhub.org/maths-for-quants/course-content/stochastic-calculus-advanced-applications/advanced-stochastic-processes/levy-processes-and-applications-in-finance-2166)
3. [The Poisson Process, in *Stochastic Processes*, Cambridge University Press](https://www.cambridge.org/core/books/abs/stochastic-processes/poisson-process/272BE8BD2D8234ACE499E8EEFFBB2A59)
4. [Càdlàg Function, Androma](https://androma.org/page/C%C3%A0dl%C3%A0g%20Function)
5. [Cadlag Modifications, Almost Sure (almostsuremath.com)](https://almostsuremath.com/2009/12/18/cadlag-modifications/)
6. [Spring 2021 A643 lecture notes, UC Berkeley](https://math.berkeley.edu/~ktawri/Notes.pdf)
7. [A. V. Skorokhod, "Limit Theorems for Stochastic Processes", Teor. Veroyatnost. i Primenen. 1:3 (1956), 289–319](https://www.mathnet.ru/php/archive.phtml?jrnid=tvp&option_lang=eng&paperid=5003&wshow=paper)
8. [Advanced Probability lecture notes, UC Berkeley (L. Tomczak)](https://math.berkeley.edu/~ltomczak/notes/Mich2022/AdvProb_Notes.pdf)
9. [Skorokhod topologies, Mathematische Semesterberichte, Springer](https://link.springer.com/article/10.1007/s00591-023-00353-2)
10. [Cadlag sample paths, JOVANA Education Glossary](https://jovanaeducation.com/glossary/cadlag-sample-paths)
11. [Stochastic Processes notes, University of Wisconsin–Madison (T. Seppäläinen)](https://people.math.wisc.edu/~tseppalainen/courses/735/notes.pdf)
12. [Mathlib.Topology.Order.Cadlag, Lean Mathlib documentation](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Topology/Order/Cadlag.html)
13. [Basics of Stochastic Analysis, University of Wisconsin–Madison (T. Seppäläinen)](https://people.math.wisc.edu/~tseppalainen/sa-book/etusivu.html)
14. [Economics 765 course notes, McGill University (R. Davidson)](https://russell-davidson.research.mcgill.ca/e765/index.html)
15. [Markovian imprecise jump processes, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0888613X22000755)
16. [Bulletin of the Institute of Mathematics, Academia Sinica](https://www.math.sinica.edu.tw/bulletins/20131/2013106.pdf)
17. [Skorokhod topology, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Skorokhod_topology)
18. [Preprint 771, CMAP, École Polytechnique](https://www.cmap.polytechnique.fr/preprint/repository/771.pdf)
19. [A. Erreygers and J. De Bock, Countable-state stochastic processes with càdlàg sample paths, arXiv:2301.07992](https://arxiv.org/abs/2301.07992)
20. [Stochastic Processes 36-754, Carnegie Mellon University (C. Shalizi)](https://www.stat.cmu.edu/~cshalizi/754/2006/)
21. [Strong Feller semigroups and Markov processes: A counter example, arXiv:2211.14789](https://arxiv.org/abs/2211.14789)
22. [Diffusions, Markov Processes, and Martingales, Volume 2, Cambridge University Press excerpt](https://assets.cambridge.org/97805217/75939/excerpt/9780521775939_excerpt.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Markov chains and processes › Continuous-time Markov processes › Feller and general-state-space continuous-time Markov processes*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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