Brownian motion $W$. It can be written into its matrix form $$\begin{aligned}
  \int_0^TvdW(s)=\int_0^T\left(\begin{array}{ccc}
  v_{11}&\cdots& v_{1m}\\
  .& &.\\
  .& &.\\
  .& &.\\
  v_{n1}&\cdots &v_{nm}
  \end{array}
  \right)\left(\begin{array}{c}
  dW_1(s)\\
  .\\
  .\\
  .\\
  dW_m(s)
    \end{array}
    \right),   
  \end{aligned}$$ which is a $n\times 1$ matrix (column vector) whose $i^{th}$ components are given by $$\begin{aligned}
  \sum_{j=1}^m\int_0^Tv_{ij}(s, \omega)dW_j(s).
  \end{aligned}$$

**\[$n$-dimensional Itô process\]**[@Oks].

Let $W$ be an $m$- Brownain motion and $v=[v_{i,j}, 1\leq i\leq n \hspace{0.2cm} 1\leq j\leq m]$ an element of $\mathbb{L}^{n\times m}([0, t], \mathbb{R}^{n\times m})$. Let $u=(u_i)_{i=1}^n$ such that $u_i\in\mathbb{L}^2([0, T])$ pour tout $1\leq i\leq n$.

The $n$-dimensional Itô process is any stochastic process of the form $$\begin{aligned}
  dX(t)=udt+vdW(t),
  \end{aligned}$$ which is a system of $n$ Itô process, where the $i^{th}$ process is given by : $$\begin{aligned}
  dX_i(t)=u_idt+\sum_{j=1}^mv_{ij}dW_j(t).
  \end{aligned}$$

**\[General Itô formula\]**

Let $  dX(t)=udt+vdW(t)$ be an $n$-dimensional Itô process. Let $g(t, x)=(g_1(t,x),..., g_p(t,x))$ be a function once differentiable with respect to $t$ and twice differentiable with respect to $x$.

Then the process $Y(t)=g(t, X(t))$ is also a $p$-dimensional Itô process, whose component $Y_k$ are given by : $$\begin{aligned}
  Y_k(t)=\dfrac{\partial g_k}{\partial t}(t, X_t)dt+\sum_{i=1}^n\dfrac{\partial g_k}{\partial x_i}(t, X_t)dX_i+\dfrac{1}{2}+\sum_{i=1}^n\sum_{j=1}^n\dfrac{\partial^2 g_k}{\partial x_i\partial x_j}(t, X_t)dX_idX_j,
  \end{aligned}$$ where $dW_idW_j=\delta_{ij}dt$ and $dW_idt=dtdW_i=0$.

[@Oks].

Stochastic process with jumps and Stochastic integral with jumps
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