to be right continuous.

A stochastic process is a family of vector random variables $(X_t)_{t\geq 0}$. That is for all $t> 0$, the application

$\begin{array}{cccc}
X_t & : \Omega &\longrightarrow &\mathbb{R}^n\\
& w & \longmapsto& X_t(\omega) 
\end{array}$ is measurable.

If $(X_t)_{t\geq 0}$ is a stochastic process, then for all $t\geq 0$, the application $t\longmapsto X_t$ is called sample path.

Let $(\mathcal{F}_t)$ be a filtration on $(\Omega, \mathcal{F}, \mathbb{P})$. A stochastic process $(X_t)$ is said to be $\mathcal{F}_t$-adapted if $\forall\hspace{0.2cm} t\geq 0$ $X_t$ is $\mathcal{F}_t$-measurable.

**\[Martingale\]** Let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration on $(\Omega, \mathcal{F}, \mathbb{P})$. A stochastic process $(M_t)_{t\geq 0}$ is called $\mathcal{F}_t$- martingale if the following properties holds

1.  $(M_t)$ is $\mathcal{F}_t$-adapted.

2.  $\mathbb{E}||M_t||<\infty $, $\forall\hspace{0.1cm} t\geq 0$.

3.  $\mathbb{E}(M_t/\mathcal{F}_s)=M_s$, $\forall\hspace{0.1cm} 0\leq s\leq t$.

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1.  If the condition $(iii)$ of the previous definition is replaced by $\mathbb{E}(M_t/\mathcal{F}_s)\geq M_s$, $\forall\hspace{0.1cm} 0\leq s\leq t$, then $(M_t)$ is called submartingale.

2.  If the condition $(iii)$ of the previous definition is replaced by $\mathbb{E}(M_t/\mathcal{F}_s)\leq M_s$, $\forall\hspace{0.1cm} 0\leq s\leq t$, then $(M_t)$ is called supermartingale.

3.  A positive submartingale is a submartingale $(X_t)_{t\geq 0}$ satisfying $X_t\geq 0$ for all $t\geq 0$.

**\[Predictable process\]**

Let $(\mathcal{F}_t)_{t\geq 0}$ be a filtration on $(\Omega, \mathcal{F}, 