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Stochastic measure distortions induced by quantile processes for risk quantification and valuation

We develop a novel stochastic valuation and premium calculation principle based on probability measure distortions that are induced by quantile processes in continuous time. Necessary and sufficient conditions are derived under which the quantile processes satisfy first- and second-order stochastic dominance. The introduced valuation principle relies on stochastic ordering so that the valuation risk-loading, and thus risk premiums, generated by the measure distortion is an ordered parametric family. The quantile processes are generated by a composite map consisting of a distribution and a quantile function. The distribution function accounts for model risk in relation to the empirical distribution of the risk process, while the quantile function models the response to the risk source as perceived by, e.g., a market agent. This gives rise to a system of subjective probability measures that indexes a stochastic valuation principle susceptible to probability measure distortions. We use the Tukey-$gh$ family of quantile processes driven by Brownian motion in an example that demonstrates stochastic ordering. We consider the conditional expectation under the distorted measure as a member of the time-consistent class of dynamic valuation principles, and extend it to the setting where the driving risk process is multivariate. This requires the introduction of a copula function in the composite map for the construction of quantile processes, which presents another new element in the risk quantification and modelling framework based on probability measure distortions induced by quantile processes.