cadlag sample paths
/ cad-LAG (kah-deh-lah-zhay) /
Cadlag is the path regularity that lets jump processes be handled rigorously: a function that is right-continuous and has left limits everywhere. The acronym is French, continue a droite, limite a gauche. It is the canonical state space of paths for Lévy processes, semimartingales, and Markov jump processes, because it is the weakest regularity that admits jumps while still pinning down a well-defined value at every time.
A function t -> x(t) is cadlag if for every t the right limit x(t+) exists and equals x(t) (right-continuity) and the left limit x(t-) exists (but may differ from x(t)). The jump at t is Delta x(t) = x(t) - x(t-), and a cadlag function on a compact interval has at most countably many jumps and only finitely many of size exceeding any epsilon > 0. For Lévy processes, stochastic continuity guarantees a cadlag version (a theorem, not an assumption), so one always works with the cadlag modification. The natural topology on cadlag path space is the Skorokhod J1 topology, which allows small time-deformations so that nearby jumps at nearby times are close — the uniform topology is too rigid because it would make X_{t} and X_{t + epsilon} far apart across a jump.
Cadlag is the reason the value of a Lévy process at a jump time is the post-jump value, and it makes the jump times into well-defined stopping times. The honest subtlety: convergence in the Skorokhod topology is not the same as uniform convergence, and a sequence of continuous paths converging in J1 can have a discontinuous limit — this is exactly how a rescaled random walk (continuous, piecewise linear) can converge to a process with jumps. Treating cadlag convergence as uniform is a common and serious error in limit theorems on path space.
A Poisson path is the prototypical cadlag function: it sits at value k, then jumps up to k+1 at an arrival time t, with N(t-) = k and N(t) = k+1. The value AT the jump is the new value, and the left limit records the old one. By contrast 1_{[0, t)} (left-continuous) is not cadlag — its value at t is not its right limit.
The Poisson path is right-continuous; its left limit holds the pre-jump value.
Cadlag is the version, not an extra hypothesis: stochastic continuity of a Lévy process already guarantees a cadlag modification exists. But the Skorokhod topology, not the uniform topology, is what makes functional limit theorems for jump processes work.