For the purpose of this question a local vol model is a 1d SDE which specifies the price process and we have a contingent claim that depends on those prices (in general, at multiple times). e.g. $dX_t = \sigma(X_t, t)dW_t$. A stochastic vol model is an at least 2d SDE where one of the equations is for the aforementioned prices process, but the additional equations specify other variables that the price process is not independent of. e.g. $dX_t = X_t Y_t dW^1_t, dY_t = \nu Y_t dW^2_t$.
Do you agree that in general, given a stochastic vol model, there is no equivalent local vol model in the following sense: The joint density across all times of the price process $X_t$ in both models can be made the same. In other words, given a set of prices of contingent claims on $X_t$(that depend on multiple dates) a stochastic vol model determines, there is no local vol model that gives the same set of prices. If so, can you point me in the direction of a proof of this? If no proof is available, a nice counterexample will suffice. A local vol and stochastic vol model which give the same vanilla options price, but have at least one different joint density.