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. We constructed quantile diffusion processes by transforming each marginal of a given univariate diffusion process under a composite map consisting of a distribution function and quantile function, which in turn produces the marginals of the resulting quantile process.
After revision: This paper focuses on the development of a new class of diffusion processes that allows for direct interpretation of model parameters such as skewness and kurtosis. The second type are function-valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. The quantile processes are obtained by transforming the marginals of a diffusion process under a composite map consisting of a distribution function and quantile function, which in turn produces the marginals of the resulting quantile process.