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

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

Published electronically:
January 20, 2011

Full-text PDF

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. Shiryaev,*Theory of Maringales*, Nauka, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR**1022664 (90j:60046)****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.**P. Billingsley,*Convergence of Probability Measures*, John Wiley & Sons, 1968. MR**0233396 (38:1718)****4.**J. Garrido,*Weak convergence of risk processes*, Insurance and Risk Theory, Nato Anvanced Study Institute on Insurance and Risk Theory, 1985, pp. 349-360. MR**864549 (87j:60011)****5.**J. Jacod and A. N. Shiryaev,*Limit Theorems for Stochastic Processes*, Springer-Verlag, New York, 1987. MR**959133 (89k:60044)****6.**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)****7.**T. Kurtz and P. Protter,*Weak limit theorems for stochastic integrals and stochastic differential equations*, Ann. Probab.**19**(1991), no. 3, 1035-1070. MR**1112406 (92k:60130)****8.**Yu. S. Mishura and D. S. Silvestrov,*Limit theorems for stochastic Riemann-Stieltjes integrals*, Theory Stoch. Process.**10 (26)**(2004), no. 1-2, 122-140. MR**2327856 (2008g:60169)****9.**P. Protter,*Stochastic Integration and Differential Equations*, 2nd ed., Stochastic Modelling and Applied Probability, vol. 21, Springer, Berlin, 2005. MR**2273672 (2008e:60001)**

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:
https://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