càdlàg process

A càdlàg process $X$ is a stochastic process for which the paths $t\mapsto X_t$ are right-continuous with left limits everywhere, with probability one. The word càdlàg is an acronym from the French for “continu à droite, limites à gauche”.Such processes are widely used in the theory of noncontinuous stochastic processes. For example, semimartingales are càdlàg, and continuous-time martingales and many types of Markov processes have càdlàg modifications.

Given a càdlàg process $X_t$ with time index $t$ ranging over the nonnegative real numbers, its left limits are often denoted by

for every $t>0$. Also, the jump at time $t$ is written as

$$\mathrm{\Delta }{X}_t=X_t-X_{t-}.$$

Alternative terms used to refer to a càdlàg process are rcll (right-continuous with left limits), R-process and right-process.

Although used less frequently, a process whose paths are almost surely left-continuous with right limits everywhere are known as càglàd, lcrl or L-processes.