TITLE: Product Rule for Ito Processes
QUESTION [14 upvotes]: Is it true that if $X(t)$ is an Ito process and $p(t)$ is non-stochastic, then the ordinary chain rule applies, that is,
$$d(X(t)p(t)) = dX(t)p(t) + X(t)p'(t)dt?$$

REPLY [9 votes]: This can be obtained directly from Ito's product rule:
$$
d(X(t)Y(t)) = X(t)dY(t) + Y(t)dX(t) + dX(t)dY(t)
$$
For illustration, assume your $dX(t)$ and $dp(t)$ has form:
$$
d(X_t) = \mu_1dt + \sigma_1dW_t \\
d(Y_t) = \mu_2dt
$$
Since your $p(t)$ is non-stochastic, it only has derivative w.r.t time and thus the final term is :
$$
dX(t)dY(t) = \mu_1dt(\mu_2dt) + \sigma_1dW_t(\mu_2dt) = 0 
$$
as a result of cross term and quadratic variation of time.
One other way to understand the product rule is to use the two-dimensional Ito formula and let $f(t,x,y) = xy$.