Suppose a stock follows the stochastic differential equation
$$dS=\frac{P-S}{\omega}dt+SdW_t,$$
such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value of an option is $V(t,S_t)$, what is the value of $dV(t,S_t)$ given by Itô's Lemma (i.e. similar to the stochastic derivation of the Black-Scholes formula)? Any help would be much appreciated.
So far I have,
$$dV=\left(\frac{\partial V}{\partial t}+\frac{P-S}{\omega}\frac{\partial V}{\partial S}+\frac{S^2}{2}\frac{\partial^2 V}{\partial S^2}\right)dt+S\frac{\partial V}{\partial S} dW_t,$$
but I'm not sure whether this is correct or how best to simplify.