201 research outputs found
An analysis of the R\"uschendorf transform - with a view towards Sklar's Theorem
In many applications including financial risk measurement, copulas have shown
to be a powerful building block to reflect multivariate dependence between
several random variables including the mapping of tail dependencies.
A famous key result in this field is Sklar's Theorem. Meanwhile, there exist
several approaches to prove Sklar's Theorem in its full generality. An elegant
probabilistic proof was provided by L. R\"{u}schendorf. To this end he
implemented a certain "distributional transform" which naturally transforms an
arbitrary distribution function to a flexible parameter-dependent function
which exhibits exactly the same jump size as .
By using some real analysis and measure theory only (without involving the
use of a given probability measure) we expand into the underlying rich
structure of the distributional transform. Based on derived results from this
analysis (such as Proposition 2.5 and Theorem 2.12) including a strong and
frequent use of the right quantile function, we revisit R\"{u}schendorf's proof
of Sklar's theorem and provide some supplementing observations including a
further characterisation of distribution functions (Remark 2.3) and a strict
mathematical description of their "flat pieces" (Corollary 2.8 and Remark 2.9)
Operators with extension property and the principle of local reflexivity
Given an arbitrary -Banach ideal , we ask for geometrical
properties of this ideal which are sufficient (and necessary) to allow a
transfer of the principle of local reflexivity to this operator class
On utility-based super-replication prices of contingent claims with unbounded payoffs
Consider a financial market in which an agent trades with utility-induced
restrictions on wealth. For a utility function which satisfies the condition of
reasonable asymptotic elasticity at we prove that the utility-based
super-replication price of an unbounded (but sufficiently integrable)
contingent claim is equal to the supremum of its discounted expectations under
pricing measures with finite {\it loss-entropy}. For an agent whose utility
function is unbounded from above, the set of pricing measures with finite
loss-entropy can be slightly larger than the set of pricing measures with
finite entropy. Indeed, the former set is the closure of the latter under a
suitable weak topology.
Central to our proof is the representation of a cone of utility-based
super-replicable contingent claims as the polar cone to the set of finite
loss-entropy pricing measures. The cone is defined as the closure, under
a relevant weak topology, of the cone of all (sufficiently integrable)
contingent claims that can be dominated by a zero-financed terminal wealth.
We investigate also the natural dual of this result and show that the polar
cone to is generated by those separating measures with finite
loss-entropy. The full two-sided polarity we achieve between measures and
contingent claims yields an economic justification for the use of the cone
, and an open question
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