Functional limit theorems for stochastic integrals with applications to risk processes and to value processes of self-financing strategies in a multidimensional market. II
Authors:
Yu. S. Mishura and Yu. V. Yukhnovs’kiĭ
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 82 (2011), 87-101
MSC (2010):
Primary 60G44, 60F05, 60B12
DOI:
https://doi.org/10.1090/S0094-9000-2011-00829-0
Published electronically:
August 4, 2011
MathSciNet review:
2790485
Full-text PDF Free Access
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Additional Information
Abstract: We study sufficient conditions for the convergence of value processes of self-financial strategies in the case of a $d$-dimensional financial market with continuous time. The conditions for the weak convergence of value processes are discussed in detail for the Black–Scholes market model. We also consider the “inverse” problem for the weak convergence of risk-minimizing strategies.
References
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- Jean Jacod, Sylvie Méléard, and Philip Protter, Explicit form and robustness of martingale representations, Ann. Probab. 28 (2000), no. 4, 1747–1780. MR 1813842, DOI https://doi.org/10.1214/aop/1019160506
- Pascale Monat and Christophe Stricker, Föllmer-Schweizer decomposition and mean-variance hedging for general claims, Ann. Probab. 23 (1995), no. 2, 605–628. MR 1334163
- R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- Martin Schweizer, Option hedging for semimartingales, Stochastic Process. Appl. 37 (1991), no. 2, 339–363. MR 1102880, DOI https://doi.org/10.1016/0304-4149%2891%2990053-F
- Hans Föllmer and Martin Schweizer, Hedging of contingent claims under incomplete information, Applied stochastic analysis (London, 1989) Stochastics Monogr., vol. 5, Gordon and Breach, New York, 1991, pp. 389–414. MR 1108430
References
- 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, Teor. Imovir. Mat. Stat. 81 (2009), 114–127; English transl. in Theor. Probability and Math. Statist. 81 (2010), 131–146. MR 2667315 (2011e:60069)
- J. Jacod, S. Meleard, and P. Protter, Explicit form and robustness of martingale representations, Ann. Probab. 28 (2000), no. 4, 1747–1780. MR 1813842 (2001m:60127)
- P. Monat and C. Striker, Föllmer–Schweizer decomposition and mean-variance hedging for general claims, Ann. Probab. 23 (1995), 605–628. MR 1334163 (97m:60065)
- R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Nauka, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664 (90j:60046)
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, 1968. MR 0233396 (38:1718)
- M. Schweizer, Option hedging for semimartingales, Stoch. Process. Appl. 37 (1993), 339–363. MR 1102880 (92c:90025)
- H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information, Appl. Stoch. Anal. (M. H. A. Davis and R. J. Elliott, eds.), vol. 5, 1991, pp. 389–414. MR 1108430 (92g:90029)
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
myus@univ.kiev.ua
Yu. V. Yukhnovs’kiĭ
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 2, Kiev 03127, Ukraine
Email:
Yuhnovskiy@hq.eximb.com
Keywords:
Stochastic integrals,
functional limit theorems,
weak convergence,
semimartingales,
semifinancing strategies
Received by editor(s):
March 22, 2010
Published electronically:
August 4, 2011
Additional Notes:
The first author is indebted to the European Commission for support in the framework of the “Marie Curie Actions” program, grant PIRSES-GA-2008-230804
Article copyright:
© Copyright 2011
American Mathematical Society
