Functional limit theorems for stochastic integrals with applications to risk processes and to self-financing strategies in a multidimensional market. I
Authors:
Yu. S. Mishura, G. M. Shevchenko and Yu. V. Yukhnovs’kiĭ
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 131-146
MSC (2010):
Primary 60G44, 60F05, 60B12
DOI:
https://doi.org/10.1090/S0094-9000-2011-00815-0
Published electronically:
January 20, 2011
MathSciNet review:
2667315
Full-text PDF Free Access
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Additional Information
Abstract: We study sufficient conditions for the weak convergence of stochastic integrals with respect to processes of bounded variation, martingales, or semimartingales. A semimartingale theorem is extended to the multidimensional case. We apply a limit procedure and pass from processes of bounded variation to risk processes. An “inverse” problem for the weak convergence is also considered.
References
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- Yu. S. Mishura, Some limit theorems for stochastic integrals with respect to a martingale and their applications, Teor. Veroyatnost. Mat. Statist. 22 (1980), 104–118; English transl. in Theory Probab. Math. Statist. 22 (1981), 115–129.
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- Thomas G. Kurtz and Philip Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (1991), no. 3, 1035–1070. MR 1112406
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References
- R. Sh. Liptser and A. N. Shiryaev, Theory of Maringales, Nauka, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664 (90j:60046)
- Yu. S. Mishura, Some limit theorems for stochastic integrals with respect to a martingale and their applications, Teor. Veroyatnost. Mat. Statist. 22 (1980), 104–118; English transl. in Theory Probab. Math. Statist. 22 (1981), 115–129.
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- P. Protter, Stochastic Integration and Differential Equations, 2nd ed., Stochastic Modelling and Applied Probability, vol. 21, Springer, Berlin, 2005. MR 2273672 (2008e:60001)
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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
G. M. Shevchenko
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:
zhora@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
Received by editor(s):
July 10, 2009
Published electronically:
January 20, 2011
Additional Notes:
The first two authors are grateful to the European Commissions for support in the framework of the program “Marie Curie Actions”, grant PIRSES-GA-2008-230804
Article copyright:
© Copyright 2011
American Mathematical Society
