The Annals of Probability

Convergence in various topologies for stochastic integrals driven by semimartingales

Adam Jakubowski

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Abstract

We generalize existing limit theory for stochastic integrals driven by semimartingales and with left-continuous integrands. Joint Skorohod convergence is replaced with joint finite dimensional convergence plus an assumption excluding the case when oscillations of the integrand appear immediately before oscillations of the integrator. Integrands may converge in a very weak topology. It is also proved that convergence of integrators implies convergence of stochastic integrals with respect to the same topology.

Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 2141-2153.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903222

Digital Object Identifier
doi:10.1214/aop/1041903222

Mathematical Reviews number (MathSciNet)
MR1415245

Zentralblatt MATH identifier
0871.60028

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60H05: Stochastic integrals 60B10: Convergence of probability measures

Keywords
Stochastic integral Skorohod topology functional convergence semimartingales

Citation

Jakubowski, Adam. Convergence in various topologies for stochastic integrals driven by semimartingales. Ann. Probab. 24 (1996), no. 4, 2141--2153. doi:10.1214/aop/1041903222. https://projecteuclid.org/euclid.aop/1041903222


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References

  • Avram, F. and Taqqu, M. (1992). Weak convergence of sums of moving averages in the -stable domain of attraction. Ann. Probab. 20 483-503.
  • Dellacherie, C. and Meyer, P. A. (1980). Probabilit´es et Potentiel 2. Hermann, Paris.
  • Jakubowski, A. (1994). A non-Skorohod topology on the Skorohod space. Unpublished manuscript.
  • Jakubowski, A., M´emin, J. and Pag es, G. (1989). Convergence en loi des suites d'int´egrales stochastiques sur l'espace D1 de Skorokhod. Probab. Theory Related Fields 81 111-137.
  • Janicki, A. and Weron, A. (1994). Simulation and Chaotic Behaviour of -Stable Stochastic Processes. Marcel Dekker, New York.
  • Kurtz, T. (1991). Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19 1010-1034.
  • Kurtz, T. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070.
  • M´emin, J. and SLomi ´nski, L. (1991). Condition UT et stabilit´e en loi des solutions d'equations diff´erentieles stochastiques. S´eminaires de Probabilit´es XXV. Lecture Notes in Math. 1485 162-177. Springer, Berlin.
  • Meyer, P. A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincar´e Probab. Statist. 20 353-372.
  • Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261-290.
  • SLomi ´nski, L. (1989). Stability of strong solutions of stochastic differential equations. Stochastic Process. Appl. 31 173-202.
  • SLomi ´nski, L. (1994). Stability of stochastic differential equations driven by general semimartingales. Unpublished manuscript.
  • Stricker, C. (1985). Lois de semimartingales et crit eres de compacit´e. S´eminaires de Probabilit´es XIX. Lecture Notes in Math. 1123. Springer, Berlin.
  • Topsøe, F. (1969). A criterion for weak convergence of measures with an application to convergence of measures on D 0 1. Math. Scand. 25 97-104.