Before revision: This paper focuses on the development of a new class of diffusion processes that allows for direct and dynamic modelling of quantile diffusions.
After revision: We propose a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: The first, which we largely focus on, features a random quantile level and allows for direct and dynamic modelling of quantile diffusions.