152 research outputs found
Recent progress in random metric theory and its applications to conditional risk measures
The purpose of this paper is to give a selective survey on recent progress in
random metric theory and its applications to conditional risk measures. This
paper includes eight sections. Section 1 is a longer introduction, which gives
a brief introduction to random metric theory, risk measures and conditional
risk measures. Section 2 gives the central framework in random metric theory,
topological structures, important examples, the notions of a random conjugate
space and the Hahn-Banach theorems for random linear functionals. Section 3
gives several important representation theorems for random conjugate spaces.
Section 4 gives characterizations for a complete random normed module to be
random reflexive. Section 5 gives hyperplane separation theorems currently
available in random locally convex modules. Section 6 gives the theory of
random duality with respect to the locally convex topology and in
particular a characterization for a locally convex module to be
prebarreled. Section 7 gives some basic results on convex
analysis together with some applications to conditional risk measures. Finally,
Section 8 is devoted to extensions of conditional convex risk measures, which
shows that every representable type of conditional convex risk
measure and every continuous type of convex conditional risk measure
() can be extended to an type
of lower semicontinuous conditional convex risk measure and an
type of continuous
conditional convex risk measure (), respectively.Comment: 37 page
The Future of Radio Revisited: Expert Perspectives and Future Scenarios for Radio Media in 2025
In 2005-2006 the research group DRACE (Digital Radio Cultures in Europe) performed a study on how 43 people in key positions related to the radio industry in four European countries and Canada viewed the future of radio and which delivery technologies they considered would be most successful. In addition, it analyzed the motives and reasons why certain technologies were seen as more promising than others. Finally, it presented four different future scenarios for radio media. The study was published in the Journal of Radio and Audio Media, May 2008. In 2005 the future of radio was considered much less obvious and clear than it appeared 10 years previously. Instead of a transition from analog to digital audio broad- casting (DAB), there was a selection of alternative technological options for digital audio delivery. When looking back from 2015 and considering the results of expert interviews, the project group about Public Service Media in the HERA project: Transnational Radio Encounters found interesting perspectives in replicating this study – now looking forward to 2025. By using the same questionnaire and interviewing the same experts (or new persons in the same positions) they could both confront the predictions with the present situation, looking for technological, regulatory, policy based, user oriented contexts. Furthermore, they could ask the experts to look ten years forward from now. Besides from the interviews, desk studies were in order to explore the national similarities and differences as background for the analysis of the scenarios for the 2015 and 2025 studies. This comparative study involves Denmark, Finland, Germany, Ireland and the UK
Multivariate risks and depth-trimmed regions
We describe a general framework for measuring risks, where the risk measure
takes values in an abstract cone. It is shown that this approach naturally
includes the classical risk measures and set-valued risk measures and yields a
natural definition of vector-valued risk measures. Several main constructions
of risk measures are described in this abstract axiomatic framework.
It is shown that the concept of depth-trimmed (or central) regions from the
multivariate statistics is closely related to the definition of risk measures.
In particular, the halfspace trimming corresponds to the Value-at-Risk, while
the zonoid trimming yields the expected shortfall. In the abstract framework,
it is shown how to establish a both-ways correspondence between risk measures
and depth-trimmed regions. It is also demonstrated how the lattice structure of
the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results
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One-sided versus two-sided stochastic descriptions
It is well-known that discrete-time finite-state Markov Chains, which are
described by one-sided conditional probabilities which describe a dependence on
the past as only dependent on the present, can also be described as
one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for
finite-spin models, which are described by two-sided conditional probabilities.
In such Markov Fields the time interpretation of past and future is being
replaced by the space interpretation of an interior volume, surrounded by an
exterior to the left and to the right.
If we relax the Markov requirement to weak dependence, that is, continuous
dependence, either on the past (generalising the Markov-Chain description) or
on the external configuration (generalising the Markov-Field description), it
turns out this equivalence breaks down, and neither class contains the other.
In one direction this result has been known for a few years, in the opposite
direction a counterexample was found recently. Our counterexample is based on
the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and
Disordered Systems
Equilibrium states and invariant measures for random dynamical systems
Random dynamical systems with countably many maps which admit countable
Markov partitions on complete metric spaces such that the resulting Markov
systems are uniformly continuous and contractive are considered. A
non-degeneracy and a consistency conditions for such systems, which admit some
proper Markov partitions of connected spaces, are introduced, and further
sufficient conditions for them are provided. It is shown that every uniformly
continuous Markov system associated with a continuous random dynamical system
is consistent if it has a dominating Markov chain. A necessary and sufficient
condition for the existence of an invariant Borel probability measure for such
a non-degenerate system with a dominating Markov chain and a finite (16) is
given. The condition is also sufficient if the non-degeneracy is weakened with
the consistency condition. A further sufficient condition for the existence of
an invariant measure for such a consistent system which involves only the
properties of the dominating Markov chain is provided. In particular, it
implies that every such a consistent system with a finite Markov partition and
a finite (16) has an invariant Borel probability measure. A bijective map
between these measures and equilibrium states associated with such a system is
established in the non-degenerate case. Some properties of the map and the
measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on
page 4 (the complete removal of the paragraph became the condition for the
publication in the DCDS-A after the reviewer ran out of the citation
suggestions collected in the paragraph
Exponential martingales and changes of measure for counting processes
We give sufficient criteria for the Dol\'eans-Dade exponential of a
stochastic integral with respect to a counting process local martingale to be a
true martingale. The criteria are adapted particularly to the case of counting
processes and are sufficiently weak to be useful and verifiable, as we
illustrate by several examples. In particular, the criteria allow for the
construction of for example nonexplosive Hawkes processes as well as counting
processes with stochastic intensities depending on diffusion processes
Representation of the penalty term of dynamic concave utilities
In this paper we will provide a representation of the penalty term of general
dynamic concave utilities (hence of dynamic convex risk measures) by applying
the theory of g-expectations.Comment: An updated version is published in Finance & Stochastics. The final
publication is available at http://www.springerlink.co
Arbitrage and deflators in illiquid markets
This paper presents a stochastic model for discrete-time trading in financial
markets where trading costs are given by convex cost functions and portfolios
are constrained by convex sets. The model does not assume the existence of a
cash account/numeraire. In addition to classical frictionless markets and
markets with transaction costs or bid-ask spreads, our framework covers markets
with nonlinear illiquidity effects for large instantaneous trades. In the
presence of nonlinearities, the classical notion of arbitrage turns out to have
two equally meaningful generalizations, a marginal and a scalable one. We study
their relations to state price deflators by analyzing two auxiliary market
models describing the local and global behavior of the cost functions and
constraints
On the dual representation of coherent risk measures
A classical result in risk measure theory states that every coherent risk measure has a dual representation as the supremum of certain expected value over a risk envelope. We study this topic in more detail. The related issues include: (1) Set operations of risk envelopes and how they change the risk measures, (2) The structure of risk envelopes of popular risk measures, (3) Aversity of risk measures and its impact to risk envelopes, and (4) A connection between risk measures in stochastic optimization and uncertainty sets in robust optimization
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