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UT condition and stability in law of stochastic differential equations. (Condition UT et stabilité en loi des solutions d’équations différentielles stochastiques.) (French) Zbl 0746.60063

Séminaire de probabilités, Lect. Notes Math. 1485, 162-177 (1991).
[For the entire collection see Zbl 0733.00018]
In [Ch. Stricker, Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 209-217 (1985; Zbl 0558.60005)] some special condition of “uniform tightness” type was introduced. Under this condition called UT a functional limit theorem for stochastic integrals has been recently proved by A. Jakubowski, J. Mémin and G. Pages [Probab. Theory Relat. Fields 81, No. 1, 111-137 (1989; Zbl 0638.60049)]. In the present paper conditions equivalent to UT are given. As a consequence the following limit theorem for solutions of stochastic differential equations is proved. Assume that \(\{ (H^ n,Z^ n)\}\) is a sequence of cadlag processes converging in distribution to \((H,Z)\) in the Skorokhod topology and that \(\{ Z^ n\}\) is a sequence of semimartingales satisfying UT. If \(f=f(x)\) is a bounded and continuous function, then a sequence of solutions of \(dX^ n = dH^ n+ f(X^ n)dZ^ n\) tends to a solution of \(dX = dH+ f(X)dZ\). Related results have been recently obtained by Th. G. Kurtz and Ph. Protter [Ann. Probab. 19, No. 3, 1035-1070 (1991)].

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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