The Annals of Applied Probability

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

Alexander Schied

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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.

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Ann. Appl. Probab., Volume 14, Number 3 (2004), 1398-1423.

First available in Project Euclid: 13 July 2004

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Zentralblatt MATH identifier

Primary: 91B28 91B30: Risk theory, insurance 62G10: Hypothesis testing

Neyman–Pearson problem robust utility functional law-invariant risk measure optimal contingent claim generalized moment problem


Schied, Alexander. On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals. Ann. Appl. Probab. 14 (2004), no. 3, 1398--1423. doi:10.1214/105051604000000341.

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