Convergence of the maximum probability of success in the problem of quantile hedging for a model of an asset price process with long-range dependence
Authors:
M. V. Bratyk, Yu. V. Kozachenko and Yu. S. Mishura
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 84 (2012), 15-31
MSC (2010):
Primary 60G22, 91B24; Secondary 60G15
DOI:
https://doi.org/10.1090/S0094-9000-2012-00861-2
Published electronically:
July 26, 2012
MathSciNet review:
2857413
Full-text PDF Free Access
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Additional Information
Abstract: The convergence in probability of the sets of maximum probability of success is studied in the problem of quantile hedging for a model of an asset price process involving Brownian and fractional Brownian motions.
References
- M. Bratyk and Y. Mishura, Quantile hedging with rediscounting on complete financial market, Prykladna statystyka. Aktuarna i finansova matematyka (2007), no. 2, 46–57.
- Patrick Cheridito, Regularizing fractional Brownian motion with a view towards stock price modelling, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Dr.sc.math.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland). MR 2715456
- Masuyuki Hitsuda, Representation of Gaussian processes equivalent to Wiener process, Osaka Math. J. 5 (1968), 299–312. MR 243614
- Hans Föllmer and Peter Leukert, Quantile hedging, Finance Stoch. 3 (1999), no. 3, 251–273. MR 1842286, DOI https://doi.org/10.1007/s007800050062
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
- M. A. Lifshits, Gaussian random functions, Mathematics and its Applications, vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736
- Alexander Melnikov and Yuliya Mishura, On pricing and hedging in financial markets with long-range dependence, Math. Financ. Econ. 5 (2011), no. 1, 29–46. MR 2810792, DOI https://doi.org/10.1007/s11579-011-0048-z
- Yu. S. Mīshura, G. M. Shevchenko, and Yu. V. Yukhnovs′kiĭ, Functional limit theorems for stochastic integrals with applications to risk processes and to capital of self-financing strategies in a multidimensional market. I, Teor. Ĭmovīr. Mat. Stat. 81 (2009), 114–127 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 81 (2010), 131–146. MR 2667315, DOI https://doi.org/10.1090/S0094-9000-2011-00815-0
- Yu. V. Kozachenko and Yu. S. Mīshura, Maximum upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations that have fractional Brownian motion with Hurst index $H<1/2$. I, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 45–56 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 51–64. MR 2321180, DOI https://doi.org/10.1090/S0094-9000-08-00713-8
References
- M. Bratyk and Y. Mishura, Quantile hedging with rediscounting on complete financial market, Prykladna statystyka. Aktuarna i finansova matematyka (2007), no. 2, 46–57.
- P. Cheridito, Regularizing Fractional Brownian Motion with a View towards Stock Price Modeling, Ph.D. Thesis, Zurich, 2001, pp. 157–173. MR 2715456
- M. Hitsuda, Representation of Gaussian processes equivalent to Wiener process, Osaka J. Math. (1968), no. 5, 299–312. MR 0243614 (39:4935)
- H. Föllmer and P. Leukert, Quantile hedging, Finance Stochast. (1999), no. 3, 251–273. MR 1842286 (2002g:91096)
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TViMS, Kiev, 1998; English transl., American Mathematical Society, Providence, Rhode Island, 2000. MR 1743716 (2001g:60089)
- M. A. Lifshits, Gaussian Random Functions, TViMS, Kiev, 1995; English transl., Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736 (98k:60059)
- A. Melnikov and Yu. Mishura, On pricing and hedging in financial markets with long-range dependence, Math. Financ. Econ. 5 (2011), no. 1, 29–46. MR 2810792 (2012f:60141)
- Yu. S. Mishura, G. M. Shevchenko, and Yu. V. Yukhnovs’kiĭ, Functional limit theorems for stochastic integrals with applications to risk processes and to self-financing strategies in a multidimensional market. I, Theor. Imovirnost. Matem. Statyst. 81 (2009), 114–127; English transl. in Theor. Probability and Math. Statist. 81 (2010), 131–146. MR 2667315 (2011e:60069)
- Yu. V. Kozachenko and Yu. S. Mishura, Maximal upper bounds for the moments of stochastic integrals and solutions of stochastic differential equations with respect to fractional Brownian motion with Hurst index $H<1/2$. I, Theor. Imovirnost. Matem. Statyst. 75 (2006), 45–59; English transl. in Theor. Probability and Math. Statist. 75 (2007), 51–64. MR 2321180 (2008g:60167)
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Additional Information
M. V. Bratyk
Affiliation:
Department of Mathematics, Faculty for Informatics, National University “Kyiv Mohyla Academy”, Skovorody Street 2, Kyiv 04070, Ukraine
Email:
mbratyk@ukr.net
Yu. V. Kozachenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine
Email:
ykoz@univ.kiev.ua
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 4E, Kiev 03127, Ukraine
Email:
myus@univ.kiev.ua
Keywords:
Quantile hedging,
fractional Brownian motion,
mixed model,
limit theorems,
probability of success
Received by editor(s):
December 24, 2010
Published electronically:
July 26, 2012
Additional Notes:
The second author is supported by the grant #230804 “Multifractionality” of the European Commission
Article copyright:
© Copyright 2012
American Mathematical Society
