Taylor Approximations for Stochastic Partial Differential Equations

Abstract

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence. In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hlder continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right. The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix. Audience: Applied and pure mathematicians interested in using and further developing numerical methods for SPDEs will find this book helpful. It may also be used as a source of material for a graduate course. Contents: Preface; List of Figures; Chapter 1: Introduction; Part I: Random and Stochastic Ordinary Partial Differential Equations; Chapter 2: RODEs; Chapter 3: SODEs; Chapter 4: SODEs with Nonstandard Assumptions; Part II: Stochastic Partial Differential Equations; Chapter 5: SPDEs; Chapter 6: Numerical Methods for SPDEs; Chapter 7: Taylor Approximations for SPDEs with Additive Noise; Chapter 8: Taylor Approximations for SPDEs with Multiplicative Noise; Appendix: Regularity Estimates for SPDEs; Bibliography; Index.