Quantile Diffusions

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 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 and a quantile function. Such maps, analogous to rank transmutation maps, produce the marginals of the resulting quantile process. 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