On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients

Abstract

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.