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Lebesgue property for convex risk measures on Orlicz spaces. (English) Zbl 1258.91108

Summary: We present a robust representation theorem for monetary convex risk measures \({\rho : \mathcal{X} \rightarrow \mathbb{R}}\) such that \[ \lim_n\rho(X_n) = \rho(X)\,\text{whenever}\,(X_n)\,\text{almost\,surely\,converges\,to}\,X, \] \({|X_n| \leq Z \in \mathcal{X}, \text{ for\,all}\,n \in \mathbb{N}}\) and \({\mathcal{X}}\) is an arbitrary Orlicz space. The separable \({\langle\mathbb{L}^{1}, \mathbb{L}^{\infty}\rangle}\) case of E. Jouini, W. Schachermayer and N. Touzi [Adv. Math. Econ. 9, 49–71 (2006; Zbl 1198.46028)], as well as the non-separable version of F. Delbaen [in: Optimality and risk – modern trends in mathematical finance. The Kabanov Festschrift. Berlin: Springer. 39-48 (2009; Zbl 1187.91095)] are contained as a particular case here. We answer a natural question posed by Biagini and Fritelli [2]. Our approach is based on the study for unbounded sets, as the epigraph of a given penalty function associated with \(\rho\), of the celebrated weak compactness Theorem due to R. C. James [Isr. J. Math. 13, 289–300 (1972; Zbl 0252.46012)].

MSC:

91B30 Risk theory, insurance (MSC2010)
91G10 Portfolio theory
91G80 Financial applications of other theories
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B50 Compactness in Banach (or normed) spaces
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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References:

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