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We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaens representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuokas representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi{c} and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
In this paper, we show that, on classical model spaces including Orlicz spaces, every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every point whose negative part has a thin tail.
In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our
When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the statistical
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