Given a mixture model of two local volatility models, the price for an option is given by:
$$V(K,T) = p V_{loc1}(K,T) + (1-p) V_{loc2}(K,T)$$
where $V_{loc}(K,T)$ is the price of the option given a dupire local volatility function and $p$ a weight.
Is there a way to retrieve the implied density for a path dependent option using this model? I am looking for a general solution which could be used on barrier options, double barriers and touch-type options.
Thanks