We develop 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 dynamic random quantile level that allows for direct interpretation of the resulting quantile process characteristics such as location, scale, skewness, and kurtosis, in terms of the model parameters. 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. By the proposed innovative and simple---yet powerful---construction method, 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 transmutation maps, produce the marginals of the resulting quantile process. Sub-classes of quantile diffusions are explored, with emphasis placed on the Tukey family of models whereby skewness and kurtosis are directly parameterised and thus the composite map is explicable with respect to such statistical behaviours. As an example of an application of quantile diffusions, we show how probability measure distortions, a form of dynamic tilting, can be induced. Though particularly useful in financial mathematics and actuarial science, examples of which are given in this work, measure distortions feature prominently across multiple research areas. For instance, dynamic distributional approximations (statistics), non-parametric and asymptotic analysis (mathematical statistics), dynamic risk measures (econometrics), behavioural economics, decision making (operations research), signal processing (information theory), and not least in general risk theory including applications thereof.

Bio: Andrea Macrina holds a PhD in Mathematics from King's College, University of London, and an MSc in Physics from the University of Bern, Switzerland. He is a Reader in Mathematics and the Director of the Financial Mathematics MSc Programme in the Department of Mathematics, University College London. Andrea holds an Adjunct Professorship at the University of Cape Town in the African Institute of Financial Markets and Risk Management.

At present, Andrea is the holder of a Fields Research Fellowship.

Past appointments include a Lectureship in Financial Mathematics in the Department of Mathematics, King's College London, a five-year Adjunct Associate Professorship in the Department of Actuarial Science, University of Cape Town, a one-year Visiting Research Associate Professorship in the Institute of Economic Research, Kyoto University, and a six-month Research Fellowship at ETH Zurich.

Andrea is one of the principle developers of information-based asset pricing, and delivers seminars and invited talks at academics conferences and to industry professionals. He is an Associate Editor of the International Journal of Theoretical and Applied Finance (IJTAF) and co-founder of the annual Financial Mathematics Team Challenge (FMTC) held in Cape Town and Rio de Janeiro.