Testing hypotheses for measures with different masses: Four optimization problems
Authors:
A. A. Gushchin and S. S. Leshchenko
Journal:
Theor. Probability and Math. Statist. 101 (2020), 109-117
MSC (2020):
Primary 62F03, 62G10
DOI:
https://doi.org/10.1090/tpms/1115
Published electronically:
January 5, 2021
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Additional Information
Abstract: We consider a problem similar to testing two composite hypotheses, where measures constituting the hypotheses are not probabilities and may have different masses. Then it is naturally to consider four different optimization problems. To characterize optimal solutions we introduce corresponding dual optimization problems. Our main goal is to find sufficient conditions for the existence of saddle points in each problem.
References
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References
- V. Baumann, Eine parameterfreie Theorie der ungünstigsten Verteilungen für das Testen von Hypothesen, Z. Wahrscheinlichkeitstheorie verw. Gebiete 11 (1968), no. 1, 41–60. MR 242328
- J. Cvitanić and I. Karatzas, Generalized Neyman–Pearson lemma via convex duality, Bernoulli 7 (2001), no. 1, 79–97. MR 1811745
- F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, Springer, Berlin, Heidelberg, 2006. MR 2200584
- H. Föllmer and P. Leukert, Quantile hedging, Finance & Stochastics 3 (1999), no. 3, 251–273. MR 1842286
- H. Föllmer and P. Leukert, Efficient hedging: Cost versus shortfall risk, Finance & Stochastics 4 (2000), no. 2, 117–146. MR 1780323
- A. A. Gushchin, A characterization of maximin tests for two composite hypotheses, Math. Meth. Statist. 24 (2015), no. 2, 110–121. MR 3366948
- A. A. Gushchin and E. Mordecki, Bounds on option prices for semimartingale market models, Proc. Steklov Inst. Math. 237 (2002), 73–113. MR 1976509
- D. Hernández-Hernández and E. Trevino-Aguilar, Efficient hedging of European options with robust convex loss functionals: A dual-representation formula, Math. Finance 21 (2011), no. 1, 99–115. MR 2779871
- O. Krafft and H. Witting, Optimale Tests und ungünstigste Verteilungen, Z. Wahrscheinlichkeitstheorie verw. Gebiete 7 (1967), no. 4, 289–302. MR 217929
- E. L. Lehmann, Testing Statistical Hypotheses, Wiley, New York, 1959. MR 0107933
- A. Melnikov and A. Nosrati, Equity-Linked Life Insurance: Partial Hedging Methods, Chapman & Hall/CRC, Boca Raton, 2017. MR 3702066
- Y. Nakano, Minimizing coherent risk measures of shortfall in discrete-time models under cone constraints, Appl. Math. Finance 10 (2003), no. 2, 163–181.
- Y. Nakano, Efficient hedging with coherent risk measure, J. Math. Anal. Appl. 293 (2004), no. 1, 345–354. MR 2052551
- Y. Nakano, Partial hedging for defaultable claims, Adv. Math. Economics 14 (2011), 127–145. MR 3220854
- A. A. Novikov, Hedging of options with a given probability, Theory Probab Appl. 43 (1999), no. 1, 135–143. MR 1670004
- B. Rudloff, Convex hedging in incomplete markets, Appl. Math. Finance 14 (2007), no. 5, 437–452. MR 2378981
- B. Rudloff, Coherent hedging in incomplete markets, Quant. Finance 9 (2009), no. 2, 197–206. MR 2512990
- R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966), no. 1, 81–89. MR 187062
- E. Treviño Aguilar, Robust efficient hedging for American options: The existence of worst case probability measures, Statistics & Decisions 27 (2009), no. 1, 1–23. MR 2597423
- E. Treviño Aguilar, Duality in a problem of static partial hedging under convex constraints, SIAM J. Financial Math. 6 (2015), 1152–1170. MR 3429733
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Additional Information
A. A. Gushchin
Affiliation:
Steklov Mathematical Institute and Lomonosov Moscow State University, Moscow, Russia
Email:
gushchin@mi-ras.ru
S. S. Leshchenko
Affiliation:
Lomonosov Moscow State University, Moscow, Russia
Email:
sslsystemup@yandex.ru
Keywords:
Convex duality,
testing hypotheses,
saddle point
Received by editor(s):
October 3, 2019
Published electronically:
January 5, 2021
Article copyright:
© Copyright 2020
American Mathematical Society
