  We develope a perturbation theory for stochastic differential equations
(SDEs) by which we mean both stochastic ordinary differential equations (SODEs)
and stochastic partial differential equations (SPDEs). In particular, we
estimate the $ L^p $-distance between the solution process of an SDE and an
arbitrary It\^o process, which we view as a perturbation of the solution
process of the SDE, by the $ L^q $-distances of the differences of the local
characteristics for suitable $ p, q > 0 $. As application of our perturbation
theory, we establish strong convergence rates for numerical approximations of a
class of SODEs with non-globally monotone coefficients. As another application
of our perturbation theory, we prove strong convergence rates for spectral
Galerkin approximations of solutions of semilinear SPDEs with non-globally
monotone nonlinearities including Cahn-Hilliard-Cook type equations and
stochastic Burgers equations. Further applications of the perturbation theory
include the regularity of solutions of SDEs with respect to the initial values
and small-noise analysis for ordinary and partial differential equations.
