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'c\`adl\`ag process'
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| Title of object: |
c\`adl\`ag process |
| Canonical Name: |
CadlagProcess |
| Type: |
Definition |
| Created on: |
2008-12-13 17:09:03 |
| Modified on: |
2008-12-13 17:36:40 |
| Classification: |
msc:60G07 |
| Keywords: |
stochastic process |
| Defines: |
c\`adl\`ag, rcll, R-process, right-process, c\`agl\`ad, lcrl, L-process |
| Synonyms: |
c\`adl\`ag process=rcll process
c\`adl\`ag process=R-process
c\`adl\`ag process=right-process |
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Content:
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A c\`adl\`ag 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 \emph{c\`adl\`ag} is an acronym from the French for ``continu \`a droite, limites \`a gauche''.
Such processes are widely used in the theory of noncontinuous stochastic processes. For example, semimartingales are c\`adl\`ag, and continuous-time martingales and many types of Markov processes have c\`adl\`ag modifications.
Given a c\`adl\`ag process $X_t$ with time index $t$ ranging over the nonnegative real numbers, its left limits are often denoted by
\begin{equation*}
X_{t-}=\lim_{\substack{s\rightarrow t,\\ s<t}}X_s
\end{equation*}
for every $t>0$. Also, the jump at time $t$ is written as
\begin{equation*}
\Delta X_t = X_t-X_{t-}.
\end{equation*}
Alternative terms used to refer to a c\`adl\`ag process are \emph{rcll} (right-continuous with left limits), \emph{R-process} and \emph{right-process}.
Although used less frequently, a process whose paths are almost surely left-continuous with right limits everywhere are known as \emph{c\`agl\`ad}, \emph{lcrl} or \emph{L-processes}.
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