## 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)

### Citations:

Zbl 0655.60026; Zbl 0733.00018; Zbl 0558.60005; Zbl 0638.60049
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