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

For nonlinear stochastic equations dx(t) = [Ax(t) + f (t,x(t), lambda)] dt + g(t, x(t), lambda) d omega(t) with parameter lambda 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 xi. 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 lambda.