Описание
vi Preface
tools for this purpose are the distributional transform, the quantile transform, and
their multivariate extensions. These tools give an easy access to the Frechet class, ´
which is a synonym for the class of all possible dependence models. In particular
a simple application of these distributional transforms gives the general form of
Sklar’s representation theorem and thus the notion of copula. The multivariate
quantile transform yields a construction method for random vectors with specified
general distribution and is a basic tool for simulation. The multivariate distributional
transform on the other side, transforms a random vector to a vector with iid
uniformly distributed components. This transform extends a classical result of
Rosenblatt (1952) and has some important applications to goodness of fit tests and
to identification procedures. Some concrete classes of constructions of multivariate
copula models are described by methods like L2-projections or the pair-copula
constructions.
A classical topic in the analysis of risk is the development of sharp risk bounds
in dependence models. The historical origins of this question are the Hoeffding–
Frechet bounds which give sharp upper and lower bounds for the covariance and the ´
joint distribution function of two random variables X, Y with distribution functions
F , G. These results have been extended to the problem of establishing sharp bounds
for general risk functionals w.r.t. Frechet classes. Important progress on this class ´
of problems was obtained by the development of a corresponding duality theory,
which was motivated by this problem of getting bounds for dependence functionals.
It turned out that this duality theory in case of a two-fold product space connects up
with the Monge–Kantorovich mass-transportation theory which aimed to describe
minimal distances or transport costs between two distributions. By means of duality
theory several basic sharp dependence bounds could be determined.
As a consequence the notion of comonotonicity is identified as the worst case
dependence structure, in case the components of the portfolio are real. These
findings were further extended by means of various stochastic ordering results
concerning diffusion type orderings (as convex order or stop-loss order) and
also concerning dependence orderings (like super-modular or directionally-convex
ordering). W.r.t. all convex law invariant risk measures comonotonicity is the worst
case dependence structure of the joint portfolio.
An exposition of the representation theory of convex risk measures and its basic
properties like continuity properties is given on spaces of Lp
-risks. Also extensions
to risk measures on portfolio vectors are detailed. These extensions allow one
to include for optimal allocation or portfolio problems the important aspect of
dependence within the portfolio components. A fundamental question concerning
the dependence structure is on the existence and form of a worst case dependence
structure – generalizing comonotonicity – for a sample X1;:;Xn of portfolio
vectors. It turns out however that a universally worst case dependence structure
does not exist any more in higher dimension. But it is possible to describe worst
case dependent portfolios w.r.t. specific multivariate risk measures. Here again a
close connection with mass transportation comes into play. The max-correlation
risk measures which are defined via mass transportation problems are the building
Детали
- Год издания
- 2013
- Format