4.  For all $t>0$, $N_t$ follows a poisson law with parameter $\lambda t$ (and we denote $N_t\leftrightsquigarrow \mathcal{P}(\lambda t)$). That is $$\begin{aligned}
     \mathbb{P}(N_t=k)=e^{-\lambda t}\dfrac{(\lambda t)^k}{k!},\hspace{0.5cm} k\in \mathbb{N}.
     \end{aligned}$$

**\[Compound poisson process\]**

Let $(Z_n)$ be a sequence of discrete independent identically distributed random variables with probability law $\nu_Z$. Let $N=(N_t)$ be a poisson process with parameter $\lambda$. Let’s assume that $(N_t)$ and $(Z_n)$ are independent. A compound poisson process with intensity $\lambda>0$ with a jump law $\nu_Z$ is a $\mathcal{F}_t$- adapted stochastic process $(Y_t)$ defined by : $$\begin{aligned}
 Y_t: =\sum_{k=1}^{N_t}Z_k.
 \end{aligned}$$

**\[Compensated poisson process\]**

A compensated poisson process associated to a poisson process $N$ with intensity $\lambda$ is a stochastic process $\overline{N}$ defined by : $$\begin{aligned}
\overline{N}(t) := N(t)-\lambda t.\end{aligned}$$

\[ch1quadratic\] Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\mathcal{F}_t)$ a filtration on this space.

If $(N)_t$ is a $\mathcal{F}_t$- adapted poisson process with intensity $\lambda$, then

1.  $\overline{N}$ is a $\mathcal{F}_t$- adapted martingale.

2.  $\mathbb{E}(\overline{N}(t+s)-\overline{N}(t))=0$.

3.  $\mathbb{E}[\overline{N}(t+s)-\overline{N}(t)]^2=\lambda s, \hspace{0.5cm} \forall\; t,s \geq 0$.

4.  $\overline{N}_t^2-\lambda t$ is a martingale.

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1.  Let $\leq s\leq t$, then $$\begin{aligned}
    \mathbb{E}(\overline{N}_t/\mathcal{F}_s)&=& \mathbb{E}(\overline{N}_t-\overline{N}_s+\overline{N}_s/ \mathcal{F}_s)\\
    &=&\mathbb{E}(\overline{N}_t-\overline{N}_s/\mathcal{F}_s)+\overline{N}_s\\
    &=&\mathbb{E}(N_t-N_s-\lambda t+\lambda s/\mathcal{F}_s)+N_s-\lambda s\\
    &=&\mathbb{E}(N_t-N_s)-\lambda t+\lambda s +N_s-\lambda s \hspace{0.1cm}\\
    & & \text{since the increments of the 