Setup
Let
$$ f=f(t,z) $$depend on time explicitly, and let
$$ dZ_t=X_t\,dt+Y_t\,dB_t. $$Formula
Then
$$ df(t,Z_t)
\frac{\partial f}{\partial t}(t,Z_t),dt + \frac{\partial f}{\partial z}(t,Z_t),dZ_t + \frac12 \frac{\partial^2 f}{\partial z^2}(t,Z_t),d\langle Z\rangle_t. $$
After substitution,
$$ df(t,Z_t)
\left( \frac{\partial f}{\partial t} + \frac{\partial f}{\partial z}X_t + \frac12 \frac{\partial^2 f}{\partial z^2}Y_t^2 \right)dt + \frac{\partial f}{\partial z}Y_t,dB_t. $$
Use
This version is the standard tool for producing time-dependent martingales.