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Dual representation of quasi-convex conditional maps. (English) Zbl 1232.46067

Let us consider a filtered probability space \((\Omega, {\mathcal F}, ({\mathcal F}_t)_{t\geq 0},\mathbb{P})\) and a map \(\pi:L_t\to L_s\) between the subspaces \(L_t\subseteq L^0(\Omega, {\mathcal F}_t,\mathbb{P})\) and \(L_s\subseteq L^0(\Omega,{\mathcal F}_s,\mathbb{P}).\) The aim of the paper is to analyse a dual representation for quasi–convex maps of this kind, following the line of the result due to [M. Volle, “Duality for the level sum of quasiconvex functions and applications”, ESAIM, Control Optim. Calc. Var. 3, 329-343 (1998; Zbl 0904.49023)] where a real–valued, lower semicontinuous function \(f:L\to \mathbb{R}\cup \{-\infty\}\cup\{+\infty\}\) was considered. Indeed, he showed that the function \(f\) can be recovered as follows: \[ f(X)=\sup_{X'\in L'}\inf_{\xi\in L}\{f(\xi)|X'(\xi)\geq X'(X)\}\tag{1} \] In Theorems 2.9 and 2.10, the main results of the paper, the authors provide a conditional version of (1): \[ \pi(X)=\mathrm{ess}\,\sup_{Q\in L_t^*\cap {\mathcal P}}R(E_Q[X|{\mathcal F}_s],Q),\tag{2} \] where \[ R(Y,Q):=\mathrm{ess}\,\inf_{\xi\in L_t}\{\pi(\xi)|\,E_Q[\xi|{\mathcal F}_s]\geq_Q Y\}, \] \(L_t^*\hookrightarrow L_t^1\) is the order continuous dual space of \(L_t,\) and \[ {\mathcal P}:= \{{dQ\over dP}|\, Q\ll \mathbb{P}\;\mathrm{and}\;Q\;\mathrm{probability}\}=\{\xi'\in L_+^1|E_P[\xi']=1\}. \] In Theorem 2.9, \(\pi\) is assumed to be lower semicontinuous with respect to the weak topology \(\sigma(L_t,L_t^*),\) while in Theorem 2.10 strong upper semicontinuity assumptions are required. Their proofs rely on the application of (1) to the real valued quasi–convex map \(\pi_A:L_t\to \overline{\mathbb{R}}\) defined by \(\pi_A(X):=\sum_{A\in \Gamma}\pi_A(X){\mathbf 1}_A,\) where \(\Gamma\) is a finite partition of \(\Omega\) of \({\mathcal F}_s\)-measurable sets \(A\in \Gamma.\)
The paper is organized as follows. In Section 2 the key definitions are provided, and the main results are presented. In particular, in Proposition 2.13, it is showed that \(\pi\) is quasi–convex, monotone, continuous from below, and regular if and only if \(\pi\) can be represented as in (2), with \(R\) belonging to a suitable class of maps. Section 3 is devoted to some preliminary results concerning the properties of the functions involved in the dual representation. Theorems 2.9 and 2.10 are proved in Section 4. In the appendix the technical important lemmas are proved.
Reviewer: Rita Pini (Milano)

MSC:

46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A20 Duality theory for topological vector spaces
91G80 Financial applications of other theories
60H99 Stochastic analysis

Citations:

Zbl 0904.49023
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