Nonlinear stochastic dynamical systems as ordinary stochastic differential
equations and stochastic difference equations are in the center of this presentation
in view of the asymptotic behavior of their moments. We study the exponential p-th
mean growth behavior of their solutions as integration time tends to infinity. For this
purpose, the concepts of attractivity, stability and contractivity exponents for moments
are introduced as generalizations of well-known moment Lyapunov exponents
of linear systems. Under appropriate monotonicity assumptions we gain uniform
estimates of these exponents from above and below. Eventually, these concepts are
generalized to describe the exponential growth behavior along certain Lyapunov-type
functionals.