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

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **84** (2011).

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

Abstract | References | Similar Articles | 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.

**1.**M. Bratyk and Y. Mishura,*Quantile hedging with rediscounting on complete financial market*, Prykladna statystyka. Aktuarna i finansova matematyka (2007), no. 2, 46-57.**2.**P. Cheridito,*Regularizing Fractional Brownian Motion with a View towards Stock Price Modeling*, Ph.D. Thesis, Zurich, 2001, pp. 157-173. MR**2715456****3.**M. Hitsuda,*Representation of Gaussian processes equivalent to Wiener process*, Osaka J. Math. (1968), no. 5, 299-312. MR**0243614 (39:4935)****4.**H. Föllmer and P. Leukert,*Quantile hedging*, Finance Stochast. (1999), no. 3, 251-273. MR**1842286 (2002g:91096)****5.**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)****6.**M. A. Lifshits,*Gaussian Random Functions*, TViMS, Kiev, 1995; English transl., Kluwer Academic Publishers, Dordrecht, 1995. MR**1472736 (98k:60059)****7.**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)****8.**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)****9.**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 . 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

DOI:
https://doi.org/10.1090/S0094-9000-2012-00861-2

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