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

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **81** (2010).

Journal:
Theor. Probability and Math. Statist. **81** (2010), 131-146

MSC (2010):
Primary 60G44, 60F05, 60B12

Published electronically:
January 20, 2011

Full-text PDF Free Access

Abstract | References | Similar Articles | 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.

**1.**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****2.**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.**3.**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****4.**J. Garrido,*Weak convergence of risk processes*, Insurance and risk theory (Maratea, 1985) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 171, Reidel, Dordrecht, 1986, pp. 349–360. MR**864549****5.**Jean Jacod and Albert N. Shiryaev,*Limit theorems for stochastic processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR**959133****6.**Jean Jacod, Sylvie Méléard, and Philip Protter,*Explicit form and robustness of martingale representations*, Ann. Probab.**28**(2000), no. 4, 1747–1780. MR**1813842**, 10.1214/aop/1019160506**7.**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****8.**Yuliya S. Mishura and Dmitrii S. Silvestrov,*Limit theorems for stochastic Riemann-Stieltjes integrals*, Theory Stoch. Process.**10**(2004), no. 1-2, 122–140. MR**2327856****9.**Philip E. Protter,*Stochastic integration and differential equations*, Stochastic Modelling and Applied Probability, vol. 21, Springer-Verlag, Berlin, 2005. Second edition. Version 2.1; Corrected third printing. MR**2273672**

Retrieve articles in *Theory of Probability and Mathematical Statistics*
with MSC (2010):
60G44,
60F05,
60B12

Retrieve articles in all journals with MSC (2010): 60G44, 60F05, 60B12

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

DOI:
http://dx.doi.org/10.1090/S0094-9000-2011-00815-0

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