Before revision:  The transformation allows for the moments of the underlying process to be directly interpreted with regard to parameters of the transformation. For instance, skewness or kurtosis may be introduced to enable more realistic modelling of data such as financial asset returns, as well as the recycling of samples of the underlying process to make simulation of the transformed quantile process easier. We derive the stochastic differential equation satisfied by the quantile diffusion, and characterise the conditions under which strong and weak solutions exist, both in the general case and for the more specific Tukey g-h, g-transform and h-transform families of quantile diffusions .
After revision:  We discuss the relationship and differences between our approach and existing methods and characterisations of quantile processes in discrete and continuous time. As an example of an application of quantile diffusions, we show how measure distortions, often found in financial mathematics and actuarial science, may be induced .