I have a question about Ito's formula for Brownian motion. If we have the following diffusion equation $$ dX_t=u(t,X_t)dt+v(t,X_t)dB_t $$
Consider the inner product of $X_t$, which is $\sum_{i=1}^n X_i^2(t)$. So what is $$ d<X, X>_t=d\left(\sum_{i=1}^n X_i^2(t)\right)=? $$
I know Ito's formula for multi-dimensional B.M. that is for $Y_t=g(t, X_t)$, then $$ dX_t=\frac{\partial g_k}{\partial t}(t,x)dt+\sum_i \frac{\partial g_k}{\partial x_i}(t,x)dX_i+\frac{1}{2}\sum_{i,j}\frac{\partial^2 g_k}{\partial x_ix_j}dX_idX_j $$