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 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.

41 pages

References in corpus (5)

• Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
• Finite Element Approximation of the Cahn-Hilliard-Cook equation
• Global flows for stochastic differential equations without global Lipschitz conditions
• Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities
• Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Cited by in corpus (15)

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• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
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• On One-Dimensional Riccati Diffusions
• $L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs
• A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations
• Strong convergence rates of an explicit scheme for stochastic Cahn--Hilliard equation with additive noise
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values
• Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise
• Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives
• Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions
• A convergent stochastic scalar auxiliary variable method
• Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients
• Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions