A function $f : [0,T]\longrightarrow \mathbb{R}^n$ is said to be right continuous with left limit at $t\in[0,T]$ if $$\begin{aligned}
     f(t^+) : =\lim_{s\longrightarrow t^+}f(s) \hspace{0.5cm}\text{and}\hspace{0.2cm}f(t^-) : =\lim_{s\longrightarrow t^-}f(s)\hspace{0.2cm}\text{exist}\hspace{0.2cm} \text{and} \hspace{0.5cm} f(t^+)=f(t). 
     \end{aligned}$$

2.  A function $f : [0,T]\longrightarrow \mathbb{R}^n$ is said to be left continuous with right limit if $$\begin{aligned}
      f(t^+) : =\lim_{s\longrightarrow t^+}f(s) \hspace{0.5cm}\text{and}\hspace{0.2cm}f(t^-) : =\lim_{s\longrightarrow t^-}f(s)\hspace{0.2cm}\text{exist}\hspace{0.2cm} \text{and} \hspace{0.5cm} f(t^-)=f(t).\end{aligned}$$

In the litterature, the french short forms “cádlág” and “cáglád” denote respectively functions which are right continous with left limit and left continous with right limit.

-   If $f$ is right continous with left limit at $t$, then $\Delta f(t)=f(t)-f(t^-)$ is called the jump of $f$ at $t$.

-   If $f$ is left continous with right limit at $t$, then $\Delta f(t)=f(t^+)-f(t)$ is called the jump of $f$ at $t$.

A stochastic process $X=(X_t)_{t\geq 0}$ is called jump process if the sample path $s\longmapsto X_s$ is left continuous (cáglág) or right continuous (cádlág) $\forall s\geq 0$.

**\[Lévy process\]**

A stochastic process $X=\{X_t,\hspace{0.3cm} t\geq 0\}$ is a Lévy process if the following conditions are fulfilled

1.  The increments on disjoint time intervals are independent. That is for $0\leq t_0<t_1<...<t_n$ $\{X_{t_j}-X_{t_{j-1}}\hspace{0.3cm} 1\leq j\leq n\}$ are independent.

2.  The increments of sample paths 