Suppose we have d-dimensional Ito process $X_t$ with:
$dX_t = b(t, X_t)dt + \sigma(t, X_t)dW_t$,
where $W_t$ is d-dimensional standard Brownian motion.
And one-dimensional Ito process $Y_t = f(X_t)$:
$dY_t = [\partial_t f + \nabla_x f^T b + 1/2 \text{Tr}(\sigma \text{Hess}_x f \sigma)]dt + \nabla_x f \sigma dW_t,$
where $f(x) : \mathbb{R}^d \rightarrow \mathbb{R}$ and $Y_t$ is derived by Ito's formula.
Is it true that $Y_t X_t$ is still d-dimensional Ito process, that is,
$Y_tX_t = Y_0X_0 + \int_0^t X_sdY_s + \int_0^t Y_sdX_s + \int_0^t dX_sdY_s?$
And if it is true, can I get the Fokker-Planck equation from this?
(Sorry for my bad English)