I am struggling a bit with some basic stuff lately:
Consider a SV model \begin{align} dS_t &= \sigma_t S_t dW_t \\ d\sigma_t &= b(\sigma_t,t) dZ_t \end{align} with $dW_t dZ_t = 0$.
I know that zero correlation does not imply independence, and in fact $S_t$ is clearly not independent of $\sigma_t$.
However, I cannot see from the above SDEs how $\sigma_t$ can depend on $S_t$, in fact I think it doesn't.
But if $\sigma_t$ were independent of $S_t$, then the local volatility function $$ LV(K,T) := E_t [ \sigma^2_T | S_T = K] = E_t [ \sigma^2_T] $$ would not depend on $K$. But this would imply a flat local vol function which doesn't make sense.
What is wrong in my reasoning?