\mathbb{P})$. A stochastic process $(X_t)_{t\geq 0}$ is called $\mathcal{F}_t$- predictable process if for all $t> 0$, $X_t$ is measurable with respect to the $\sigma$-algebra generated by $\{X_s, \hspace{0.2cm}s<t\}$.

Let $M=(M_t)$ be a submartingale. Then for $1<p<\infty$, we have

1.  **Markov’s inequality** $$\begin{aligned}
      \mathbb{P}\left(\sup_{0\leq s \leq t}||M_t||\geq \alpha\right)\leq\dfrac{\mathbb{E}(||M_t||)}{\alpha},\hspace{0.6cm} \forall\, \alpha>0.
      \end{aligned}$$

2.  **Doop’s maximal inequality** $$\begin{aligned}
      \mathbb{E}\left[\left(\sup_{0\leq s \leq t}||M_t||\right)^p\right]^{1/p}\leq \dfrac{p}{p-1}\mathbb{E}\left[||M_t||^p\right]^{1/p}.
      \end{aligned}$$

[@Phi].

**\[Wiener process or Brownian motion\]**

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\mathcal{F}_t)_{t\geq 0}$ a filtration on this space. A $\mathcal{F}_t$-adapted stochastic process $(W_t)_{t\geq 0}$ is called Wiener process or Brownian motion if :

1.  $W_0=0$.

2.  $ t  \longmapsto W_t$ is almost surely continous.

3.  $(W_t)_{t\geq 0}$ has independent increments $($i.e $W_t-W_s$ is independent of $W_r,\hspace{0.2cm} r\leq s)$.

4.  $W_t-W_s\rightsquigarrow \mathcal{N}(0, t-s)$, for $0\leq s \leq t$. Usually, this property is called stationarity.

If $(W_t)$ is an $\mathcal{F}_t$- Brownian motion, then the following process are $\mathcal{F}_t$- martingales

1.  $W_t$.

2.  $W_t^2-t$.

3.  $\exp\left(\gamma W_t-\gamma^2\dfrac{t}{2}\right),\hspace{0.3cm}\forall\; \gamma\in \mathbb{R}$.

Let $0\leq s\leq t$, then

1.  $$\begin{aligned}
       \mathbb{E}(W_t/\mathcal{F}_s)&=&\mathbb{E}(W_t-W_s+W_s/\mathcal{F}_s)\\
       &=&W_s+\mathbb{E}(W_t-W_s/\mathcal{F}_s) \hspace{0.1cm}\text{since} \hspace{0.2cm} W_s \hspace{0.1cm} is \hspace{0.1cm} \mathcal{F}_s-\text{measurable}\\
      & =& W_s+ \mathbb{E}(W_t-W_s)\hspace{0.1cm} (\text{since the increments are independents }) \\
      &=&W_s \hspace{0.3cm}(\text{since} \hspace{0.1cm} W_t-W_s\rightsquigarrow\mathcal{N}(0,t-s)).
      \end{aligned}$$

2.  $$\begin{aligned}
      \mathbb{E}(W_t^2-t/\mathcal{F}_s)&=&\mathbb{E}(W_t^2+W_s^2-2W_sW_t+2W_sW_t-W_s^2/\mathcal{F}_s)-t\\
      &=&\mathbb{E}((W_t-W_s)^2/\mathcal{F}_s)+W_s\mathbb{E}((2W_t-W_s)/\mathcal{F}_s)-t \\
      & & 