I'm trying to establish that a function is differentiable when the value of the function follows an SDE. In a sense, I am trying to establish a converse of Ito's Lemma. More specifically, I know two pieces of information:
I know that a process $(\xi_t)_{t\ge0}$ follows an SDE $$ d \xi_t = (\delta \xi_t - X_t)dt + \omega_t dW_t, $$ where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $(X_t)_{t \ge 0}$ is some adapted process. $(\omega_t)_{t\ge 0}$ is also an adapted process.
I also know that $\xi_t = G(Y_t, V_t)$ for a continuous function $G$. Here, $(Y_t)_{t\ge 0}$ and $(V_t)_{t\ge 0}$ are two adapted processes, with volatilities $\sigma_Y(Y_t,V_t)$ and $\sigma_V(Y_t,V_t)$, respectively.
Based on this information, can I conclude that $$ \omega_t = G_Y(Y_t,V_t) \sigma_Y(Y_t,V_t) + G_V(Y_t,V_t) \sigma_V(Y_t,V_t)? $$ Or, more generally, can I conclude that the function $G$ is differentiable?