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The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$ When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any ...
A general model (with continuous paths) can be written $$\frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S$$ where the short rate $r_t$ and spot volatility $\sigma_t$ are random processes. In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions (actually constant). This produces a flat smile for any $$C(t,S;T,K) = ... 1 Here "dynamics" means the assumed future behaviour of the spot process, namely that it follows the SDE$$ dS/S = r dt + \sigma_{loc}(S,t) dW_t . There are various ways to see that these dynamics are unrealistic. One is to look for time homogeneity. In normal cases, you expect the market to follow the same rules in one week and in one year from today. ...