Càdlàg

Càdlàg is an acronym from the French continue à droite, limite à gauche, used to describe a class of real-valued or path-valued functions in probability theory. A function f defined on [0, ∞) is càdlàg if it is right-continuous and has finite left limits at every t > 0. More precisely, for all t ≥ 0, lim_{h↓0} f(t+h) = f(t), and for all t > 0, the limit lim_{s↑t} f(s) exists. The left limit is usually written f(t−).

The left limit may differ from the value at the jump time, which captures a discontinuity of the path. At t = 0, left limits are not required. Càdlàg paths are central in the Skorokhod space D([0, ∞), E), the space of càdlàg functions taking values in a metric space E, equipped with a topology used to study convergence of stochastic processes.

In stochastic processes, many models have càdlàg sample paths. Poisson processes have càdlàg paths with jumps of size 1 at random times; compound Poisson processes are similar with jump sizes given by a distribution. More generally, Lévy processes have almost surely càdlàg paths. By contrast, Brownian motion has continuous paths, which are a special case of càdlàg paths. The càdlàg property allows jumps while preserving a well-behaved right-continuous trajectory structure, making it a standard regularity condition in probability and finance, queueing theory, and related areas.