Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

2016/10/27 by Gao, Niushan, Leung, Denny H., Xanthos, Foivos
#FOS: Economics and business #Mathematical Finance (q-fin.MF)

paper · doi:10.48550/arxiv.1610.08806

Abstract

Let (Φ,Ψ) be a conjugate pair of Orlicz functions. A set in the Orlicz space L^Φ 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^Φ characterizes closedness with respect to the topology σ(L^Φ,L^Ψ). (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in L^Φ, order closedness and σ(L^Φ,L^Ψ)-closedness are indeed equivalent. In general, however, coincidence of order closedness and σ(L^Φ,L^Ψ)-closedness of convex sets in L^Φ is equivalent to the validity of the Krein-Smulian Theorem for the topology σ(L^Φ,L^Ψ); that is, a convex set is σ(L^Φ,L^Ψ)-closed if and only if it is closed with respect to the bounded-σ(L^Φ,L^Ψ) topology. As a result, we show that order closedness and σ(L^Φ,L^Ψ)-closedness of convex sets in L^Φ are equivalent if and only if either Φ or Ψ satisfies the Δ₂-condition. Using this, we prove the surprising result that: If (and only if) Φ and Ψ both fail the Δ₂-condition, then there exists a coherent risk measure on L^Φ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair (L^Φ, L^Ψ). A similar analysis is carried out for the dual pair of Orlicz hearts (H^Φ,H^Ψ).

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