I have seen $2$ "forms" of the Ito formula which are essentially, in the end, equivalent. But my question is, having seen quite a few questions on stochastic differential equations, I am wondering when one is used and when another is.
One form is
$$dY_t=\bigg(\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dX_t)^2 \bigg)$$
while another is given, using the quadratic variation $(dW_t)^2=dt$ that
$$dY_t=\bigg(\frac{\partial f}{\partial t}dt+\mu \frac{\partial f}{\partial x}+\frac{\sigma^2}{2}\frac{\partial^2 f}{\partial x^2}\bigg)dt+\sigma \frac{\partial f}{\partial x}dW_t$$
where $W_t$ is a Brownian Motion and $dX_t=\mu dt+\sigma dW_t$.
I saw some times, people use the former to find solutions and whatnot while some have used the latter.
What tells you, in a question, to use which form? Which one is more efficient to use than the other for some questions? Is there a general idea as to how to judge that?
Thank you
The second is a specialization of the first to the case $dX_t = \mu dt + \sigma dW_t$. If your $X_t$ satisfies this, use the second one, if not -- use the more generic one.