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In the present paper, given an evolving mixture of probability densities, wedefine a candidate diffusion process whose marginal law follows the sameevolution. We derive as a particular case a stochastic differential equation(SDE) admitting a unique strong solution and whose density evolves as a mixtureof Gaussian densities. We present an interesting result on the comparisonbetween the instantaneous and the terminal correlation between the obtainedprocess and its squared diffusion coefficient. As an application tomathematical finance, we construct diffusion processes whose marginal densitiesare mixtures of lognormal densities. We explain how such processes can be usedto model the market smile phenomenon. We show that the lognormal mixturedynamics is the one-dimensional diffusion version of a suitable uncertainvolatility model, and suitably reinterpret the earlier correlation result. Weexplore numerically the relationship between the future smile structures ofboth the diffusion and the uncertain volatility versions.

The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation