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January 2020 On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients
Martin Hutzenthaler, Arnulf Jentzen
Ann. Probab. 48(1): 53-93 (January 2020). DOI: 10.1214/19-AOP1345

Abstract

We develop 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 one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

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Martin Hutzenthaler. Arnulf Jentzen. "On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients." Ann. Probab. 48 (1) 53 - 93, January 2020. https://doi.org/10.1214/19-AOP1345

Information

Received: 1 September 2018; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206753
MathSciNet: MR4079431
Digital Object Identifier: 10.1214/19-AOP1345

Subjects:
Primary: 65C30

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • January 2020
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