Solutions of a stochastic differential equation forced onto a manifold by a large drift. (English) Zbl 0749.60053

Katzenberger, G. S.

Ann. Probab. 19, No. 4, 1587-1628 (1991).

It is considered a sequence of \(R^ d\)-valued semimartingales \(X_ n\) satisfying \[ X_ n(t)=X_ n(0)+\int_ 0^ t\sigma_ n(X_ n(s- ))dZ_ n(s)+\int_ 0^ t F(X_ n(s-))dA_ n(s), \] where \(Z_ n\) is a sequence of \(R^ k\)-valued semimartingales, \(\sigma_ n\) is a continuous \(d\times k\) matrix-valued function, \(F\) is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points \(\Gamma\) and \(A_ n\) is a nondecreasing process. The author investigates stability of this system. Stability means, roughly speaking, if \(X_ n(0)\) is close to \(\Gamma\) (or is only in the domain of attraction of \(\Gamma\) under the flow of \(F\)) and \(X_ n\) is close to \(\Gamma\), then any limiting process is close to \(\Gamma\). There are given conditions under which a sequence \((X_ n)\) is relatively compact in the Skorokhod topology and which ensure the existence of a stochastic integral equation which any limiting process has to satisfy.

Reviewer: J.Jakubowski (Warszawa)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

semimartingales; flow; domain of attraction; limiting process; stochastic integral equation