This thesis presents a novel approach for the construction of quantile processes, governing the stochastic dynamics of quantiles in continuous time. Two constructions are proposed, one producing a function-valued quantile process and the second, a process with random quantile levels. The latter method employs a distortion map composed of a distribution function and a quantile function, similar to a transmutation map, applied to each marginal of a 'driving' process with cadlag paths. A multidimensional extension that utilises a copula is also presented. As a result, we obtain a one-step approach to constructing widely flexible classes of stochastic models, accommodating extensive ranges of higher-order moment behaviours (e.g., tail behaviours in the finite dimensional distributions, and asymmetry). Such features are parameterised in the composite map and are thus interpretable with respect to the driving process. Sub-classes of quantile processes 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 regard to such statistical behaviours. It is also shown that the quantile processes induce a distorted probability measure that is interpretable in its properties (which may be intentionally constructed), leading to the central application developed in this thesis. We propose a general, time-consistent, and dynamic risk valuation principle under the induced measures of quantile processes, allowing for pricing in incomplete markets and thus having application in insurance pricing. Here, the distorted measures are considered 'subjective' and are constructed in such a way to account for external market characteristics and investor risk attitudes, leading to a parametric system of risk-sensitive probability measures, indexed by such factors. The properties of the valuation principle based on the quantile process distortion measures are discussed with regard to stochastic ordering and risk-loadings, and a case study is presented where insurance instruments linked to greenhouse gas emissions are considered.