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 $F$ to a flexible parameter-dependent function which exhibits exactly the same jump size as $F$. 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 $p$-Banach ideal $(0 < p \leq 1)$, 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 $-\infty$ 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 $C_U$ of utility-based super-replicable contingent claims as the polar cone to the set of finite loss-entropy pricing measures. The cone $C_U$ 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 $C_U$ 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 $C_U$, and an open question