On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally 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 Itô 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.

Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

“…It could also be regarded as one of the basic finite elements methods in spatial approximation (see e.g. [5,6,15,17,19,21,22,39]). There are also a lot of work on other spatial approximation methods including Fourier method, piecewise linear approximation, finite elements methods (see e.g.…”

“…Approximations to SPDEs driven by trace-class Wiener process have been studied a lot in the literatures (see e.g. [2,7,10,17,19,22,24,26,39]). For SPDEs driven by space-time white noise, we refer to [5,8,11,12,20,21,23] and the references therein for the convergence of spatial approximations and refer to [3,5,8,11,12,21,23] and the references therein for the convergence of temporal approximations.…”

Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients

“…By contrast, as already shown by Higham, Mao and Stuart [12], Mao and Szpruch [31], Andersson and Kruse [1], the backward (implicit) Euler method, computationally much more expensive than the explicit Euler method, can be strongly convergent under certain non-globally Lipschitz conditions. These observations suggest that special care must be taken to construct and analyze convergent numerical schemes in nonglobally Lipschitz setting, and this interesting subject has been investigated in a great portion of the literature [1,3,4,6,8,15,16,17,18,19,20,22,25,26,27,30,31,32,33,37,40,41,42,43,44,45,47,49,50,51]. In 2012, Hutzenthaler, Jentzen and Kloeden [18] introduced an explicit method, called the tamed Euler method, to numerically solve SDEs with super-linearly growing drift coefficients and globally Lipschitz diffusion coefficients.…”

“…In 2012, Hutzenthaler, Jentzen and Kloeden [18] introduced an explicit method, called the tamed Euler method, to numerically solve SDEs with super-linearly growing drift coefficients and globally Lipschitz diffusion coefficients. Since then, various explicit schemes are designed and analyzed for SDEs with (more general) locally Lipschitz coefficients [3,4,6,15,16,19,26,27,32,33,40,41,42,44,45,47,49,50,51]. Readers can, e.g., refer to [16] for a more comprehensive list of references.…”

Analysis of a Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise

“…For SPDEs with monotone coefficients, there exist fruitful results on strong error analysis of temporal and spatial numerical approximations (see, e.g., [2,5,6,17,20,21]). However, there exists only a few results in the scientific literature which establish strong and weak convergence rates for a time discrete approximation scheme in the case of an SPDE with a nonglobally monotone nonlinearity (see, e.g., [11,12,19,23,24,25]). This motives us to construct strong and weak approximations for this kind of SPDE.…”