are stationary : $X_t-X_s\approx X_{t-s}$ for $0\leq t\leq s$.

3.  The sample paths are right continuous with left limit.

The Brownian motion and the poisson process starting at $0$ are Lévy process.

[@Oksa] Let $\textbf{D}_{ucp}$ denote the space of càdlàg adapted process equipped with the topology of the uniform convergence in probability (ucp) on compact sets. ucp : $H_n\longrightarrow H$ if $\forall\; t\geq 0$ $\sup\limits_{0\leq s\leq t}|H_n(s)-H(s)|\longrightarrow 0$ in probability ($A_n\longrightarrow A$ in probability if $\forall\; \epsilon>0, \exists\; n_{\epsilon}\in \mathbb{N}$ such that $n>n_{\epsilon}\Longrightarrow \mathbb{P}(|A_n-A|>\epsilon)<\epsilon$).

In the sequel $\textbf{L}_{ucp}$ denote the space of adapted càdlàg processes (left continous with right limit) equiped with the ucp topology.

[@Oksa] Let $H$ be an elementary function. i.e there exist a partition

$0=t_0<t_1...<t_n=T$ such that $$\begin{aligned}
 H=\sum_{j=0}^nH_j1_{[t_j, t_{j+1})},
 \end{aligned}$$ where $H_j$ are $\mathcal{F}_{t_j}$-measurable. Let $(X_t)$ be a Lévy process. The stochastic integral $\int_0^tH(s)dX(s)$ is defined by $$\begin{aligned}
J_XH(t) : =\int_0^tH(s)dX(s): =\sum_{j=0}^n H_j(X(t_{j+1})-X(t_j))  \hspace{0.3cm} t\geq 0.\end{aligned}$$

[@Oksa] Let $X$ be a semimartingale, then the mapping $J_X$ can be extended to the continuous linear map $$\begin{aligned}
J_X : \textbf{L}_{ucp}\longrightarrow \textbf{D}_{ucp}.\end{aligned}$$

The above proposition allows us to define a stochastic integral of the form $$\begin{aligned}
\int_0^tH(s)dX(s),\end{aligned}$$ where $H\in \textbf{L}_{ucp}$.

[@Oksa] For $H\in \textbf{L}_{ucp}$ we define $\int_0^tH(s)dX(s)$ : $$\begin{aligned}
