Before revision:  We also introduce a multivariate uncertain volatility model of which our multivariate local volatilities model is a multivariate markovian projectionand analyze the dependence structure induced by our multivariate dynamics in detail .
After revision:  We also introduce a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model .