Volume 29, Issue 1 -

Conditional risk and acceptability mappings quantify the desirability of random variables (e.g. financial returns) by accounting for available information. In this paper the focus lies on acceptability mappings, concave translation-equivariant monotone mappings L p (Ω, F ,ℙ) →  L p ´ (Ω, F´ ,ℙ) with 1 ≤  p ´ ≤  p  ≤ ∞, where the σ -algebras F ´ ⊂  F describe the available information. Based on the order completeness of L p (Ω, F ,ℙ)-spaces, we analyze superdifferentials and concave conjugates of conditional acceptability mappings. The related results are used to show properties of two important classes of multi-period valuation functionals: SEC-functionals and additive acceptability compositions. In particular, we derive a chain rule for superdifferentials and use it for characterizing the conjugates of additive acceptability compositions and SEC-functionals.

Many statistical estimation techniques for high-dimensional or functional data are based on a preliminary dimension reduction step, which consists in projecting the sample X 1 ,..., X n onto the first D eigenvectors of the Principal Component Analysis (PCA) associated with the empirical projector ^ Π D . Classical nonparametric inference methods such as kernel density estimation or kernel regression analysis are then performed in the (usually small) D -dimensional space. However, the mathematical analysis of this data-driven dimension reduction scheme raises technical problems, due to the fact that the random variables of the projected sample (^Π D X 1 ,...,^Π D X n ) are no more independent. As a reference for further studies, we offer in this paper several results showing the asymptotic equivalencies between important kernel-related quantities based on the empirical projector and its theoretical counterpart. As an illustration, we provide an in-depth analysis of the nonparametric kernel regression case.

We consider some risk indicators of vectorial risk processes. These indicators take into account the dependencies between business lines as well as some temporal dependencies. By using stochastic algorithms, we may estimate the minimum of these risk indicators, under a fixed total capital constraint. This minimization may apply to capital reserve allocation.

The notion of asymptotic portfolio loss order is introduced to compare multivariate stochastic risk models with respect to extreme portfolio losses. In the framework of multivariate regular variation comparison criteria are derived in terms of spectral measures. This allows for analytical and numerical verification in applications. Worst and best case dependence structures with respect to the asymptotic portfolio loss order are determined. Comparison criteria in terms of further stochastic ordering notions are derived. The examples include elliptical distributions and multivariate regularly varying models with Gumbel, Archimedean, and Galambos copulas. Particular interest is paid to the inverse influence of dependence on the diversification of risks with infinite expectations.