Local stability and parameter dependence of mild solutions for stochastic differential equations

Abstract

For nonlinear stochastic equations dx(t )=( Ax(t )+ f (t, x(t), λ)) dt + g(t, x(t), λ) dω(t )w ith parameter λ in a Hilbert space, we show the existence and uniqueness of mild solutions. Provided that f satisfies a locally Lipschitz condition and g is a uniformly Lipschitz function, some sufficient conditions for p (p ≥ 2) moment locally exponential stability as well as almost surely exponential stability of mild solutions are obtained under a sufficiently small initial value ξ . Meanwhile, we also consider parameter dependence of stable mild solutions for the stochastic system if f , g are sufficiently small Lipschitz perturbations in the parameter λ.