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

Jakubowski, A.; Memin, J.; Pages, G.

Probab. Theory Relat. Fields 81, No. 1, 111-137 (1989).

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 \(X^ n\).

MSC:

60F17	Functional limit theorems; invariance principles
60H05	Stochastic integrals

Keywords:

uniform tightness; convergence of stochastic integrals; conditions on local characteristics of semimartingales; functional convergence; Skorokhod’s space

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