On Lipschitz dependence in systems with differentiated inputs (Q802210)

An extension [given by a representation of \textit{H. Doss}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 13, 99-125 (1977; Zbl 0359.60087), and \textit{H. J. Sussmann}, Ann. Probab. 6, 19-41 (1978; Zbl 0391.60056)] of the mapping \(z\to x\), \(\dot x=f(x)+G(x)\dot z\), to \(z\in L^{\infty}([0,1])\) is considered. Results on its (local) Lipschitz continuity (w.r to \(L^ p\)-norms (p\(\leq \infty)\), and Skorokhod-type metrics) as well as estimates on its local Lipschitz coefficients are obtained. These estimates are used to investigate preservation of convergence rates under the mapping \(z\to x\) in case of random inputs z which are large with small probability, and to obtain rate-of-convergence estimates of some approximate solutions of stochastic differential equations.