On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals

By Alexander Schied. 2004

93 citations recorded by the source corpus.

Motivated by optimal investment problems in mathematical finance, we consider a variational problem of Neyman-Pearson type for law-invariant robust utility functionals and convex risk measures. Explicit solutions are found for quantile-based coherent risk measures and related utility functionals. Typically, these solutions exhibit a critical phenomenon: If the capital constraint is below some critical value, then the solution will coincide with a classical solution; above this critical value, the solution is a superposition of a classical solution and a less risky or even risk-free investment. For general risk measures and utility functionals, it is shown that there exists a solution that can be written as a deterministic increasing function of the price density.

Subfields: Computational Theory and Mathematics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/895ae4ce8324d210f9228c29fe6320b59a91e77e

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In Management Science and Operations Research

Citations in the corpus

Stochastic Orders and Risk Measures: Consistency and Bounds (2006, unknown): “Therefore it applies to risk measures as considered in Schied (2004).”
Risk Measures and Robust Optimization Problems (2006, methodology): “In particular, we will give short proofs of recent results due to Sekine[66] and the author[61].”
Optimal stopping under probability distortion. (2011, background): “In particu(4)Quantile formulation has been introduced and developed in the context of financial portfolio selection involving probability distortion (see [6, 23] and [4] for earlier works).”
A REPRESENTATION RESULT FOR CONCAVE SCHUR CONCAVE FUNCTIONS (2005, methodology): “In the concave case, to show existence, one can also use standard σ (L∞( ), L1( )) upper semicontinuity arguments (see Schied 2004).”
Duality theory for optimal investments under model uncertainty (2005, background): “The same is true for lawinvariant robust utility functionals as considered in [20] and by Jouini et al.”
PORTFOLIO CHOICE VIA QUANTILES (2010, methodology): “For more recent study/survey on quantile-related performance measures see Föllmer and Schied (2004), Dhaene et al. (2006), and Cherny and Madan (2009). that these problems cannot be solved, at least directly, by conventional approaches.”
ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES (2016, unknown): “In Section 6, we discuss the implications of our results with respect to the equity premium puzzle, and 5The idea of quantile formulation was around earlier, e.g., in Schied (2004) and in Carlier and Dana (2006), who studied maximization problems with concave criteria.”
Robust projections in the class of martingale measures (2006, background): “As shown by Gilboa and Schmeidler [20], such robust preferences can be characterized by certain behavioral axioms, and they resolve several well-known “paradoxa” which arise in the classical framework; see, for instance, Karni and Schmeidler [28] or Föllmer and Schied [16], Chapter 2.5.”
On the optimal risk allocation problem (2006, background): “There exists a solution X∗ of (L) as shown in Schied (2004) if % is lower semicontinuous.”
Consistent risk measures for portfolio vectors (2006, background): “Further by an extension of Lemma 2.2 in Schied (2004) to the multivariate case we obtain that for any convex law invariant risk measure % on L∞d (P ) and any X,Y ∈ L∞d (P ) holds:”
Robust Preferences and Robust Portfolio Choice (2009, methodology): “The following result from SCHIED [2005] reduces the robust utility maximization problem to a standard utility maximization problem plus the computation of a least favorable measure, which is independent of the utility function.”