19 research outputs found
Dual Representation of Quasiconvex Conditional Maps
We provide a dual representation of quasiconvex maps between two lattices of
random variables in terms of conditional expectations. This generalizes the
dual representation of quasiconvex real valued functions and the dual
representation of conditional convex maps.Comment: Date changed Added one remark on assumption (c), page
Universal Arbitrage Aggregator in Discrete Time Markets under Uncertainty
In a model independent discrete time financial market, we discuss the
richness of the family of martingale measures in relation to different notions
of Arbitrage, generated by a class of significant sets, which we
call Arbitrage de la classe . The choice of reflects
into the intrinsic properties of the class of polar sets of martingale
measures. In particular: for S= absence of Model Independent
Arbitrage is equivalent to the existence of a martingale measure; for
being the open sets, absence of Open Arbitrage is equivalent to
the existence of full support martingale measures. These results are obtained
by adopting a technical filtration enlargement and by constructing a universal
aggregator of all arbitrage opportunities. We further introduce the notion of
market feasibility and provide its characterization via arbitrage conditions.
We conclude providing a dual representation of Open Arbitrage in terms of
weakly open sets of probability measures, which highlights the robust nature of
this concept
On Conditional Chisini Means and Risk Measures
Given a real valued functional T on the space of bounded random variables, we
investigate the problem of the existence of a conditional version of nonlinear
means. We follow a seminal idea by Chisini (1929), defining a mean as the
solution of a functional equation induced by T. We provide sufficient
conditions which guarantee the existence of a (unique) solution of a system of
infinitely many functional equations, which will provide the so called
Conditional Chisini mean. We apply our findings in characterizing the
scalarization of conditional Risk Measures, an essential tool originally
adopted by Detlefsen and Scandolo (2005) to deduce the robust dual
representation
On quasiconvex conditional maps
Quasiconvex analysis has important applications in several optimization problems in science, economics and in finance, where convexity may be lost due to absence of global risk aversion, as for example in Prospect Theor
CONDITIONAL CERTAINTY EQUIVALENT
In a dynamic framework, we study the conditional version of the classical notion of certainty equivalent when the preferences are described by a stochastic dynamic utility u(x,t,ω). We introduce an appropriate mathematical setting, namely Orlicz spaces determined by the underlying preferences and thus provide a systematic method to go beyond the case of bounded random variables. Finally we prove a conditional version of the dual representation which is a crucial prerequisite for discussing the dynamics of certainty equivalents.Stochastic dynamic utility, conditional certainty equivalent, Musielak-Orlicz spaces, quasiconcavity, dual representation
Complete duality for quasiconvex dynamic risk measures on modules of the -type
In the conditional setting we provide a complete duality between quasiconvex risk measures defined on modules of the type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.