Convergence en loi des suites d’integrales stochastiques sur l’espace \({\mathbb{D}}^ 1\) de Skorokhod. (Convergence in law of sequences of stochastic integrals on the Skorokhod space \({\mathbb{D}}^ 1)\). (French) Zbl 0638.60049

For every \(n\in N\), let us consider \((K^ n,X^ n)\), a pair of real valued cadlag processes, defined on a filtered space \((\Omega ^ n,{\mathcal F}^ n,({\mathcal F}^ n_ t),P^ n)\) with the property that \(X^ n\) is a semimartingale on this space. Let us assume that \((K^ n,X^ n)\) converges in law to (K,X), (i.e. weak convergence of the laws of processes \((K^ n,X^ n)\) on the space \({\mathbb{D}}^ 2\), a space of functions defined on \({\mathbb{R}}^ +\) with values in \({\mathbb{R}}^ 2\), endowed with Skorokhod’s topology). Let us assume also that for \((X^ n)\) holds a special property of “uniform tightness” type.
Then stochastic integral \(K_ -.X\) can be defined and the sequence \((K^ n_ -.X^ n)\) converges in law to \(K_ -.X.\)
This is the main result of the paper; included are also conditions (which are often realized) for getting “uniform tightness” property, and other auxiliary useful results: There is also a result of convergence of stochastic integrals under conditions on local characteristics of semimartingales \(X^ n\).


60F17 Functional limit theorems; invariance principles
60H05 Stochastic integrals
Full Text: DOI


[1] Billingsley, P.: Convergence of probability measures. New York: Wiley 1968 · Zbl 0172.21201
[2] Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, tome 2. Paris: Hermann 1980
[3] Follmer, H.: Calcul d’Ito sans probabilités; séminaire de probabilités XV. Lect. Notes Math., vol. 850. Berlin Heidelberg New York: Springer 1981
[4] Jacob, J.: Calcul stochastique et problèmes de martingales. Lect. Notes Math., vol. 714. Berlin Heidelberg New York: Springer 1979 · Zbl 0414.60053
[5] Jacod, J.: Théorèmes limite pour les processus; cours de l’école d’été de Saint-Flour. Lect. Notes Math., vol. 1117. Berlin Heidelberg New York: Springer 1985
[6] Jacob, J., Memin, J., Metivier, M.: On tightness and stopping times. Stoch. Proc. Appl. 14, 109-146 (1983) · Zbl 0501.60029
[7] Jakubowski, A.: On the Skorokhod topology. Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. B22, 263-285 (1986) · Zbl 0609.60005
[8] Lindvall, T.: Weak convergence of probability measures and random functions in the function Space D[0,?[. J. Appl. Probab. 10, 109-121 (1973) · Zbl 0258.60008
[9] Liptser, R.Ch., Shiryayev, A.N.: On a problem of necessary and sufficient conditions in the functional central limit theorem for local martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 312-318 (1982) · Zbl 0482.60030
[10] McLeish, D.L.: An extended martingale invariance principle. Ann. Probab. 6, 144-150 (1978) · Zbl 0379.60046
[11] Memin, J.: Théorèmes limite fonctionnels pour les processus de vraisemblance (cadre asymptotiquement non gaussien). Publications IRMAR. 1985 (Rennes 1986) · Zbl 0637.60005
[12] Meyer, P.A., Zheng, W.A.: Tightness criteria for laws of semimartingales. Ann. Inst. Henri Poincaré, Nouv. Ser., Sect. B20, 353-372 (1984) · Zbl 0551.60046
[13] Pages, G.: Un théorème de convergence fonctionnelle pour les intégrales stochastiques. Séminaires de Probabilités XX. Lect. Notes Math., vol. 11. Berlin Heidelberg New York: Springer 1986
[14] Rootzen, H.: On the functional limit theorem for martingales. Z. Wahrscheinlichkeitstheor. Verw. Geb. 51, 79-94 (1980) · Zbl 0402.60033
[15] Stominski, L.: Approximation of predictable characteristics of processes with filtrations. Preprint, Univ. Torun, Pologne, 1985
[16] Skorokhod, A.V.: Limit theorems for stochastic processes. Th. Probab. Appl. I, 423-439 (1956) · Zbl 0074.33802
[17] Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423-439 (1965) · Zbl 0135.18701
[18] Stricker, C.: Caractérisation des semimartingales. Séminaire de probabilité XVIII. Lect. Notes Math., vol. 1059. Berlin Heidelberg New York: Springer 1984
[19] Stricker, C.: Lois de semimartingales et critères de compacité. Séminaires de probabilités XIX. Lect. Notes Math., vol. 1123. Berlin Heidelberg New York: Springer 1985
[20] Yor, M.: Les inégalités de sousmartingales comme conséquences de la relation de domination. Stochastics, 3, 1-17 (1979)
[21] Avram, F.: Weak convergence of the variations, iterated integrals and Doléans-Dade exponentials of sequences of semimartingales. Ann. Probab. 16, 246-250 (1988) · Zbl 0636.60029
[22] Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987 · Zbl 0635.60021
[23] Strasser, H.: Martingale difference arrays and stochastic integrals. Probab. Th. Rel. Fields 72, 83-98 (1986) · Zbl 0575.60043
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.