Jakubowski, Adam Convergence in various topologies for stochastic integrals driven by semimartingales. (English) Zbl 0871.60028 Ann. Probab. 24, No. 4, 2141-2153 (1996). Author’s summary: We generalize existing limit theory for stochastic integrals driven by semimartingales and with left-continuous integrands. Joint Skorokhod 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. Reviewer: J.Steinebach (Marburg) Cited in 1 ReviewCited in 10 Documents MSC: 60F17 Functional limit theorems; invariance principles 60H05 Stochastic integrals 60B10 Convergence of probability measures Keywords:Skorokhod topology; functional convergence; semimartingale; stochastic integrals; Skorokhod convergence PDFBibTeX XMLCite \textit{A. Jakubowski}, Ann. Probab. 24, No. 4, 2141--2153 (1996; Zbl 0871.60028) Full Text: DOI References: [1] Avram, F. and Taqqu, M. (1992). Weak convergence of sums of moving averages in the -stable domain of attraction. Ann. Probab. 20 483-503. · Zbl 0747.60032 [2] Dellacherie, C. and Meyer, P. A. (1980). Probabilités et Potentiel 2. Hermann, Paris. · Zbl 0464.60001 [3] Jakubowski, A. (1994). A non-Skorohod topology on the Skorohod space. Unpublished manuscript. [4] Jakubowski, A., Mémin, J. and Pag es, G. (1989). Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod. Probab. Theory Related Fields 81 111-137. · Zbl 0638.60049 [5] Janicki, A. and Weron, A. (1994). Simulation and Chaotic Behaviour of -Stable Stochastic Processes. Marcel Dekker, New York. · Zbl 0946.60028 [6] Kurtz, T. (1991). Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19 1010-1034. · Zbl 0742.60036 [7] Kurtz, T. and Protter, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053 [8] Mémin, J. and SLomi ński, L. (1991). Condition UT et stabilité en loi des solutions d’equations différentieles stochastiques. Séminaires de Probabilités XXV. Lecture Notes in Math. 1485 162-177. Springer, Berlin. · Zbl 0746.60063 [9] Meyer, P. A. and Zheng, W. A. (1984). Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré Probab. Statist. 20 353-372. · Zbl 0551.60046 [10] Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1 261-290. · Zbl 0074.33802 [11] SLomi ński, L. (1989). Stability of strong solutions of stochastic differential equations. Stochastic Process. Appl. 31 173-202. · Zbl 0673.60065 [12] SLomi ński, L. (1994). Stability of stochastic differential equations driven by general semimartingales. Unpublished manuscript. [13] Stricker, C. (1985). Lois de semimartingales et crit eres de compacité. Séminaires de Probabilités XIX. Lecture Notes in Math. 1123. Springer, Berlin. · Zbl 0558.60005 [14] 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. · Zbl 0186.51501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.