# 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

 $X_{t-}=\lim_{\begin{subarray}{c}s\rightarrow t,\\ s

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

 $\Delta X_{t}=X_{t}-X_{t-}.$

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.

 Title càdlàg process Canonical name CadlagProcess Date of creation 2013-03-22 18:36:36 Last modified on 2013-03-22 18:36:36 Owner gel (22282) Last modified by gel (22282) Numerical id 7 Author gel (22282) Entry type Definition Classification msc 60G07 Synonym cadlag process Synonym rcll process Synonym R-process Synonym right-process Related topic UcpConvergenceOfProcesses Defines cadlag Defines rcll Defines R-process Defines right-process Defines càglàd Defines lcrl Defines L-process