be a complete probability space, $(W_t)_{t\in \mathbb{R}_{+}}$ a one-dimensional Brownian motion and $f :  \mathbb{R}\longrightarrow \mathbb{R}$ such that $f$ is once derivable. If $(X_t)$ is any process of the form ,

then $f(X_t)$ is an Itô processes and $$\begin{aligned}
   f(X_t)=f(X_0)+\int_0^t f'(X_s)u_sds+\dfrac{1}{2}\int_0^tf''(X_s)v_s^2ds+\int_0^tf'(X_s)v_sdW_s.
   \end{aligned}$$

[@Oksa].

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

If in the previous proposition we consider $f :[0, \infty)\times\mathbb{R}\longrightarrow\mathbb{R}$ such that $f$ is once differentiable with respect to the first variable $t$ and twice differentiable with respect to the second variable $x$,

then $f(t, X_t)$ is an Itô process and $$\begin{aligned}
   f(t, X_t)=f(0, X_0)+\int_0^t\dfrac{\partial f}{\partial t}(s,X_s)ds+\int_0^t\dfrac{\partial f}{\partial x}(s,X_s)u_sds+\int_0^t\dfrac{\partial f}{\partial x}(s,X_s)v_sdW_s+\dfrac{1}{2}\int_0^t\dfrac{\partial^2 f}{\partial x^2}(s,X_s)v_s^2ds,
   \end{aligned}$$ or in its differential form : $$\begin{aligned}
   df(t, X_t)=\dfrac{\partial f}{\partial t}(t, X_t)dt+\dfrac{\partial f}{\partial x}(t, X_t)dX_t+\dfrac{1}{2}\dfrac{\partial^2 f}{\partial x^2}(t, X_t)(dX_t)^2.
   \end{aligned}$$ $(dX_t)^2=dX_tdX_t$ is computed according to the rules $$\begin{aligned}
   dtdt=dW_tdt=dtdW_t=0,\hspace{1cm}  dW_tdW_t=dt.
   \end{aligned}$$

[@Oksa].

### Multi-dimensional Itô integral

**\[m-dimensional Brownian motion\]**[@Oks].

Let $W_1, \cdots W_m$ be $m$ Brownian motions. The random variable $W=(W_1, W_2, ..., W_m)$ is called $m$-dimensional Brownian motion. Let $\mathbb{L}^{n\times m}([0, T], \mathbb{R}^{n\times m})$ denotes the set of $n\times m$ matrices $v=[v_{ij}(t, \omega)]$, $1\leq i\leq n$, $1\leq j\leq m$. Where $v_{ij}(t, \omega)\in \mathbb{L}^2([0, T], \mathbb{R})$. $\int_0^tv_sdW_s$ denotes the Itô integral of $v$ with respect to the m-dimensional 