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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736290

Digital Object Identifier
doi:10.1214/105051604000000341

Mathematical Reviews number (MathSciNet)
MR2071428

Zentralblatt MATH identifier
1121.91054

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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1089736290


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References

  • Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • Bednarski, T. (1981). On solutions of minimax test problems for special capacities. Z. Wahrsch. Verw. Gebiete 58 397–405.
  • Cvitanić, J. (2000). Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. 38 1050–1066.
  • Cvitanić, J. and Karatzas, I. (2001). Generalized Neyman–Pearson lemma via convex duality. Bernoulli 7 79–97.
  • Dana, R.-A. and Carlier, G. (2002). Core of convex distortions of a probability on an non atomic space. Preprint, Ceremade, Univ. Paris-Dauphine.
  • Delbaen, F. (2002). Coherent measures of risk on general probability spaces. In Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann (K. Sandmann and P. J. Schönbucher, eds.) 1–37. Springer, Berlin.
  • Delbaen, F. (2000). Coherent Risk Measures. Scuola Normale Superiore, Classe di Scienze, Pisa.
  • Denneberg, D. (1994). Non-additive Measure and Integral. Kluwer Academic, Dordrecht.
  • Föllmer, H. and Leukert, P. (1999). Quantile hedging. Finance Stoch. 3 251–273.
  • Föllmer, H. and Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4 117–146.
  • Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6.
  • Föllmer, H. and Schied, A. (2002). Robust representation of convex measures of risk. In Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann (K. Sandmann and P. J. Schönbucher, eds.) 39–56. Springer, Berlin.
  • Föllmer, H. and Schied, A. (2002). Stochastic Finance: An Introduction in Discrete Time. Springer, Berlin.
  • Gilboa, I. and Schmeidler, D. (1989). Maxmin expected utility with non-unique prior. J. Math. Econ. 18 141–153.
  • Huber, P. (1981). Robust Statistics. Wiley, New York.
  • Huber, P. and Strassen, V. (1973). Minimax tests and the Neyman–Pearson lemma for capacities. Ann. Statist. 1 251–263.
  • Kirch, M. (2002). Maximin-optimal tests and least favorable pairs for concave power functions. Preprint, TU Wien.
  • Kirch, M. (2002). Efficient hedging in incomplete markets under model uncertainty. Preprint, TU Wien.
  • Kulldorff, M. (1993). Optimal control of favorable games with a time limit. SIAM J. Control Optim. 31 52–69.
  • Kusuoka, S. (2001). On law invariant coherent risk measures. Adv. Math. Econ. 3 83–95.
  • Österreicher, F. (1978). On the construction of least favourable pairs of distributions. Z. Wahrsch. Verw. Gebiete 43 49–55.
  • Pham, H. (2002). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12 143–172.
  • Rieder, H. (1977). Least favourable pairs for special capacities. Ann. Statist. 5 909–921.
  • von Weizsäcker, H. and Winkler, G. (1979). Integral representations in the set of solutions of a generalized moment problem. Math. Ann. 246 23–32.\goodbreak
  • Winkler, G. (1988). Extreme points of moment sets. Math. Oper. Res. 13 581–587.