Consider for $t \geq 0$ the following delay differential equation $$dS_t=\mu \left (\int_{-d}^0 S(t+\xi) d \xi \right ) S_t dt + \sigma \left (\int_{-d}^0 S(t+\xi) d \xi \right ) S_t dW_t$$ where $S_0=s>0, S_t=\phi(t)>0, -d \leq t \leq 0$ which represents the (known) past of $S$, $\mu, \sigma \colon \mathbb{R} \to \mathbb{R}$ are the drift and volatility which depend in a non linear way by a linear functional the past of $S$.
I can rewrite it as $$dS_t=\mu_t S_t dt + \sigma_t S_t dW_t$$ where $\mu_t=\mu \left (\int_{-d}^0 S(t+\xi) d \xi \right )$, $\sigma_t=\sigma \left (\int_{-d}^0 S(t+\xi) d \xi \right )$
Supposing that everything is smooth can I apply ito formula and get that $S_t=S_0 e^{X_t}$ where $$dX_t=(\mu_t-\sigma_t^2 /2)dt+\sigma_t dW_t, \quad X_0=log(s)$$ so that I can conclude that $S(t)>0$ since $S_0=s>0$
Is the conclusion correct? In particular can I say that $\mu_t, \sigma_t$ are well defined processes (are the Markovian?) so that $S_t$ is an Ito process and then Ito formula can be applied?