My professor asserts
However, Oksendal asserts in his textbook: $X_{t} Y_{t}-X_{0} Y_{0}=\int_{0}^{t} X_{s} d Y_{s}+\int_{0}^{t} Y_{s} d X_{s}+\int_{0}^{t}dXdY$
These are not equivalent - consider $e^{t/2}W_{t}$ for standard brownian motion $W_{t}$ - the "$g_{t}dW_{t}$" term for $e^{t/2}$ is such that $g_{t}=0$.
Is my professor's version incorrect?