variable.

**\[ Itô’s integral\]**

The Itô’s Integral of the simple process $(X_t)_{t\in\mathbb{R}}\in\mathbb{L}^2([0,T], \mathbb{R})$ is defined by $$\begin{aligned}
I_t(X)=\int_0^tX(s)dW(s) :=\sum_{j=0}^{n-1}\theta_j(W_{t_{j+1}}-W_{t_j}).\end{aligned}$$

If $f$ is an elementary function in $\mathbb{L}^2([a,b],\mathbb{R})$ and $W_t$ a Brownian motion, then :

1.  $\mathbb{E}\left(\int_a^bf(t)dW_t\right)=0$.

2.  $\mathbb{E}\left(\int_a^bf(t)dW_t\right)^2=\int_a^b\mathbb{E}(f^2(t))dt$.

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1.  By definition we have $$\begin{aligned}
    \int_a^bf(t)dW_t=\sum_{j=0}^{n-1}f_j(W_{t_j+1}-W_{t_j}).\end{aligned}$$ By taking expectation in both sides, we obtain $$\begin{aligned}
    \mathbb{E}\left[\int_a^bf(t)dW_t\right]=\sum_{j=0}^{n-1}\mathbb{E}(f_j)\mathbb{E}(W_{t_{j+1}}-W_{t_j})=0,\end{aligned}$$ since $W_{t_{j+1}}-W_{t_j}$ is a normal distribution with mean $0$ and standard deviation $\sqrt{t_{j+1}-t_j}$.

2.  $$\begin{aligned}
    \left(\int_a^bf(t)dW_t\right)^2&=&\left[\sum_{j=0}^{n-1}f_j(B_{t_{j+1}}-W_{t_j})\right]^2\\
    &=&\sum_{j=0}^{n-1}(f_j)^2(W_{t_{j+1}}-W_{t_j})^2+\sum_{l=0}^{n-1}\sum_{k=0, k\neq l}^{n-1}f_lf_k(W_{t_{l+1}}-W_{t_l})(W_{t_{k+1}}-W_{t_k}).\end{aligned}$$

    Taking expectation in both sides and using independence of the increments of Brownian motion, we get $$\begin{aligned}
    \mathbb{E}\left(\int_a^bf(t)dW_t\right)^2&=&\sum_{j=0}^{n-1}\mathbb{E}(f_j)^2E\left(W_{t_{j+1}}-W_{t_j}\right)^2\\
    &=&\sum_{j=0}^{n-1}\mathbb{E}(f_j)^2(t_{j+1}-t_j)\\
    &=&\int_a^b\mathbb{E}(f^2(t))dt.\end{aligned}$$

The following proposition can be seen in [@Oks].

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 all $t\geq 0$, there exist a sequence $(f^{(n)}_t)_{t\geq 0}$ of simple process such that $\mathbb{E}|f^{(n)}_t|^2<\infty$ and $$\begin{aligned}
\lim_{n\longrightarrow \infty}\mathbb{E}\left[\int_0^t|X_s-f_s^{(n)}|^2ds\right]=0.\end{aligned}$$

For any process $X=(X_t)_{t\geq 0}\in\mathbb{M}^2([0,T], \mathbb{R})$, we define a stochastic integral of $X$ with respect to a Brownian motion $W$ by : $$\begin{aligned}
 \int_0^tX_sdW(s)=\lim_{n\longrightarrow \infty}\int_0^tf^{(n)}_sdW(s),
 \end{aligned}$$ where $(f^{(n)}_t)$ is the sequence of simple process converging almost surely to $X$ according to the previous proposition. Moreover, using Itô isometry for elementaries functions one can prove that the limit on this definition does not depend on the actual 