  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.
