[Paper Review] Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

Niushan Gao, Denny H. Leung+1 more|arXiv (Cornell University)|Oct 27, 2016

Risk and Portfolio Optimization22 references16 citations

TL;DR

This paper establishes that for norm-bounded convex sets in Orlicz spaces $L^\Phi$, order closedness (closed under a.s. convergence) is equivalent to $\sigma(L^\Phi, L^\Psi)$-closedness if and only if at least one of the conjugate Orlicz functions $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition. The key contribution is proving that when both $\Phi$ and $\Psi$ fail the $\Delta_2$-condition, there exist coherent risk measures with the Fatou property that lack dual representation via the dual pair $(L^\Phi, L^\Psi)$, resolving a long-standing open problem in financial mathematics.

ABSTRACT

Let $(\\Phi,\\Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^\\Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^\\Phi$ characterizes closedness with respect to the topology $\\sigma(L^\\Phi,L^\\Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^\\Phi$, order closedness and $\\sigma(L^\\Phi,L^\\Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $\\sigma(L^\\Phi,L^\\Psi)$-closedness of convex sets in $L^\\Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $\\sigma(L^\\Phi,L^\\Psi)$; that is, a convex set is $\\sigma(L^\\Phi,L^\\Psi)$-closed if and only if it is closed with respect to the bounded-$\\sigma(L^\\Phi,L^\\Psi)$ topology. As a result, we show that order closedness and $\\sigma(L^\\Phi,L^\\Psi)$-closedness of convex sets in $L^\\Phi$ are equivalent if and only if either $\\Phi$ or $\\Psi$ satisfies the $\\Delta_2$-condition. Using this, we prove the surprising result that: \\emph{If (and only if) $\\Phi$ and $\\Psi$ both fail the $\\Delta_2$-condition, then there exists a coherent risk measure on $L^\\Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^\\Phi, L^\\Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^\\Phi,H^\\Psi)$.

Motivation & Objective

• To resolve the open problem of whether order closedness implies $\sigma(L^\Phi, L^\Psi)$-closedness for convex sets in Orlicz spaces $L^\Phi$.
• To characterize when the Krein-Smulian theorem holds for the topology $\sigma(L^\Phi, L^\Psi)$, linking it to the $\Delta_2$-condition on the Orlicz functions.
• To investigate the implications for dual representation of coherent risk measures in Orlicz and Orlicz heart spaces.
• To demonstrate the existence of coherent risk measures with the Fatou property that fail Fenchel-Moreau dual representation when both $\Phi$ and $\Psi$ fail the $\Delta_2$-condition.

Proposed method

• Uses duality theory in Banach function spaces, particularly the pairing between Orlicz spaces $L^\Phi$ and their associate spaces $L^\Psi$.
• Applies the Krein-Smulian theorem in the context of $\sigma(L^\Phi, L^\Psi)$-topology to characterize closedness of convex sets.
• Employs sequence analysis and dominated convergence techniques to study order closedness in $L^\Phi$ and $H^\Phi$, focusing on norm-bounded sequences converging a.s.
• Constructs explicit counterexamples using sequences of random variables in $H^\Phi$ to show that $\sigma(H^\Phi, H^\Psi)$-closedness does not imply order closedness when $\Phi$ and $\Psi$ fail $\Delta_2$.
• Analyzes the structure of representing sets in dual pairs via coordinate-wise convergence and norm convergence in $c_0$ and $\ell^1$ to verify membership in convex sets.
• Utilizes the Fenchel-Moreau duality theorem to relate lower semicontinuity and dual representation of risk measures.

Experimental results

Research questions

• RQ1 Under what conditions on the Orlicz functions $\Phi$ and $\Psi$ is order closedness equivalent to $\sigma(L^\Phi, L^\Psi)$-closedness for convex sets in $L^\Phi$?
• RQ2 Is the Krein-Smulian theorem valid for the topology $\sigma(L^\Phi, L^\Psi)$ if and only if at least one of $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition?
• RQ3 Can a coherent risk measure on $L^\Phi$ have the Fatou property without admitting a dual representation via the dual pair $(L^\Phi, L^\Psi)$?
• RQ4 What is the relationship between the $\Delta_2$-condition and the existence of a dual representation for risk measures in Orlicz spaces?
• RQ5 Does the failure of both $\Phi$ and $\Psi$ to satisfy the $\Delta_2$-condition imply the existence of a coherent risk measure with the Fatou property that lacks a dual representation?

Key findings

• Order closedness and $\sigma(L^\Phi, L^\Psi)$-closedness of convex sets in $L^\Phi$ are equivalent if and only if at least one of $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition.
• The equivalence of order closedness and $\sigma(L^\Phi, L^\Psi)$-closedness holds for norm-bounded convex sets in $L^\Phi$ if and only if the Krein-Smulian theorem is valid for $\sigma(L^\Phi, L^\Psi)$.
• When both $\Phi$ and $\Psi$ fail the $\Delta_2$-condition, there exists a coherent risk measure on $L^\Phi$ that satisfies the Fatou property but does not admit a Fenchel-Moreau dual representation with respect to the dual pair $(L^\Phi, L^\Psi)$.
• A similar failure of dual representation occurs in the Orlicz heart space setting: if both $\Phi$ and $\Psi$ fail the $\Delta_2$-condition, then there exists a coherent risk measure on $H^\Phi$ with the Fatou property that lacks dual representation via $(H^\Phi, H^\Psi)$.
• The proof constructs a sequence $X_{sr}$ in $H^\Phi$ converging in $\sigma(H^\Phi, H^\Psi)$-topology to $-W_0$, which is not in the original convex set $\mathcal{C}$, demonstrating that $\mathcal{C}$ is not $\sigma(H^\Phi, H^\Psi)$-closed.
• The construction relies on coordinate-wise convergence in $c_0$ and $\ell^1$, norm boundedness in $H^\Phi$, and the use of projections $\mathcal{P}_j$ to control the behavior of sequences in the dual space.