Ito's product rule in higher dimension I'm looking for an analogue version of Ito's famous product rule in higher dimensions. Meaning, let X, Y be $d$-dimensional (Ito-)processes. Then something similar to the following should hold:
$
\left\langle X_{t},Y_{t}\right\rangle =\left\langle X_{0},Y_{0}\right\rangle +\int_{0}^{t}\left\langle X_{s-},dY_{s}\right\rangle +\int_{0}^{t}\left\langle Y_{s-},dX_{s}\right\rangle +\left\langle dX,dY\right\rangle _{s}$
Especially I'm interested in how to interpret the quadratic co-variation term. 
Maybe I just fail at giving a perfect notation for this, please help :)
Edit: Maybe this is more appealing.
$
X_{t}^{\mathsf{T}}Y_{t}=X_{0}^{\mathsf{T}}Y_{0}+\int_{0}^{t}X_{s-}^{\mathsf{T}}dY_{s}+\int_{0}^{t}Y_{s-}^{\mathsf{T}}dX_{s}+\left[X,Y\right]_{t}$

You may use the dot product notation, that is,
\begin{align*}
X_t \cdot Y_t &= \sum_{i=1}^d X_t^i Y_t^i \\
&=\sum_{i=1}^d\left[X_0^i Y_0^i + \int_0^t X_{t-}^i dY_t^i + \int_0^t Y_{t-}^i dX_t^i + [X^i, Y^i]_t\right]\\
&=X_0\cdot Y_0 + \int_0^t X_{t-} \cdot dY_t + \int_0^t Y_{t-} \cdot dX_t + [X, Y]_t,
\end{align*}
where $$[X, Y]_t = \sum_{i=1}^d [X^i, Y^i]_t.$$
