choice of $(f^{(n)}).$

**\[Properties of Itô integral\]**.

For any process $X=(X_t)_{t\geq 0}\in\mathbb{M}^2([0,T], \mathbb{R})$ such that $\mathbb{E}|X_t|^2<\infty$, for any functions

$f,g\in \mathbb{M}^2([0,T], \mathbb{R})$ and $0\leq S<U<T$, the following holds :

1.  $\int_S^TfdW(t)=\int_S^UfdW(t)+\int_U^TfdW(t)$ almost surely.

2.  $\int_S^T(cf+g)dW(t)=c\int_S^TfdW(t)+\int_S^TgdW(t)$, for any constant $c$.

3.  $\int_S^TfdW(t)$ is $\mathcal{F}_T$-measurable.

4.  $\mathbb{E}\left(\int_0^tX_sdW(s)\right)=0$.

5.  $\mathbb{E}\left(\int_0^tX_sdW(s)\right)^2=\int_0^t\mathbb{E}(X_s^2)ds$.

[@Oks]

[@Oks] For any elementary function $f^{(n)}$ $\mathcal{F}_t$-adapted, the integral $$\begin{aligned}
 I_n(t, \omega)=\int_0^tf^{(n)}dW(r)
 \end{aligned}$$ is a martingale with respect to $\mathcal{F}_t$.

For $t\leq s$, we have : $$\begin{aligned}
 \mathbb{E}[I_n(s,\omega)/\mathcal{F}_t]&=&\mathbb{E}\left[\left(\int_0^sf^{(n)}dW(r)\right)/\mathcal{F}_t\right]\\
 &=&\mathbb{E}\left[\left(\int_0^tf^{(n)}dW(r)\right)/\mathcal{F}_t\right]+
 \mathbb{E}\left[\left(\int_t^sf^{(n)}dW(r)\right)/\mathcal{F}_t\right]\\
 &=&\int_0^tf^{(n)}dW(r)+\mathbb{E}\left[\sum_{t\leq t^{(n)}_j\leq t^{(n)}_{j+1}\leq s}f^{(n)}_j\Delta W_j/\mathcal{F}_t\right]\\
 &=&\int_0^tf^{(n)}dW(r)+\sum_{t\leq t^{(n)}_j\leq t^{(n)}_{j+1}\leq s}\mathbb{E}[f^{(n)}_j\Delta W_j/\mathcal{F}_t]\\
  &=&\int_0^tf^{(n)}dW(r)+\sum_{t\leq t^{(n)}_j\leq t^{(n)}_{j+1}\leq s}\mathbb{E}[\mathbb{E}[f^{(n)}_j\Delta W_j/\mathcal{F}_{t_j}]/\mathcal{F}_t]\\
  &=&\int_0^tf^{(n)}dW(r)+\sum_{t\leq t^{(n)}_j\leq t^{(n)}_{j+1}\leq s}\mathbb{E}[f^{(n)}_j\mathbb{E}[\Delta W_j/\mathcal{F}_{t_j}]/\mathcal{F}_t]\\
  &=&\int_0^tf^{(n)}dW(r),\hspace{0.2cm} \text{since} \hspace{0.2cm} E[\Delta W_j/\mathcal{F}_{t_j}]=\mathbb{E}[\Delta W_j]=0\\
  &=&I_n(t, \omega).
 \end{aligned}$$

**\[Generalisation\]**

Let $f(t, \omega)\in \mathbb{M}^2([0, T], \mathbb{R})$ for all $t$. Then the integral $$\begin{aligned}
   M_t(\omega)=\int_0^tf(s, \omega)dW(s)
   \end{aligned}$$ is a martingale with respect to $\mathcal{F}_t$ and $$\begin{aligned}
   \mathbb{P}\left[\sup_{0\leq t\leq T}|M_t|\geq \lambda\right]\leq\dfrac{1}{\lambda^2}\mathbb{E}\left[\int_0^Tf^2(s, \omega)ds\right], \hspace{0.3cm} \forall\; \lambda >0
   \end{aligned}$$

[@Oks].

### One dimensional Itô Formula

**\[1-dimensional Itô process\]**

Let $W_t$ be a $1$-dimensional Brownian motion on $(\Omega, \mathcal{F}, \mathbb{P})$. An Itô process (or Stochastic integral) is any stochastic process $X_t$ of the form $$\begin{aligned}
   X_t=X_0+\int_0^tu(s,\omega)ds+\int_0^tv(s,  \omega)dW(s),
   \label{ch1Ito1}
   \end{aligned}$$ where $u\in \mathbb{L}^1([0,T], \mathbb{R})$ and $v\in\mathbb{L}^2([0, T], \mathbb{R})$.

**\[ first $1$- dimensional Itô formula\]**

Let $(\Omega, \mathcal{F}, \mathbb{P})$ 