56 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
On the super replication price of unbounded claims
In an incomplete market the price of a claim f in general cannot be uniquely
identified by no arbitrage arguments. However, the ``classical'' super
replication price is a sensible indicator of the (maximum selling) value of the
claim. When f satisfies certain pointwise conditions (e.g., f is bounded from
below), the super replication price is equal to sup_QE_Q[f], where Q varies on
the whole set of pricing measures. Unfortunately, this price is often too high:
a typical situation is here discussed in the examples. We thus define the less
expensive weak super replication price and we relax the requirements on f by
asking just for ``enough'' integrability conditions. By building up a proper
duality theory, we show its economic meaning and its relation with the
investor's preferences. Indeed, it turns out that the weak super replication
price of f coincides with sup_{Q\in M_{\Phi}}E_Q[f], where M_{\Phi} is the
class of pricing measures with finite generalized entropy (i.e., E[\Phi
(\frac{dQ}{dP})]<\infty) and where \Phi is the convex conjugate of the utility
function of the investor.Comment: Published at http://dx.doi.org/10.1214/105051604000000459 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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
A unified framework for utility maximization problems: An Orlicz space approach
We consider a stochastic financial incomplete market where the price
processes are described by a vector-valued semimartingale that is possibly
nonlocally bounded. We face the classical problem of utility maximization from
terminal wealth, with utility functions that are finite-valued over
, , and satisfy weak regularity
assumptions. We adopt a class of trading strategies that allows for stochastic
integrals that are not necessarily bounded from below. The embedding of the
utility maximization problem in Orlicz spaces permits us to formulate the
problem in a unified way for both the cases and .
By duality methods, we prove the existence of solutions to the primal and dual
problems and show that a singular component in the pricing functionals may also
occur with utility functions finite on the entire real line.Comment: Published in at http://dx.doi.org/10.1214/07-AAP469 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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