New York: Springer, 2017. — 391 p.Contents:Prologue Why random and Brownian motion (white noise)? What color is the noise? Solutions Modeling with SPDEs Specific topics of this book Brownian motion and stochastic calculus Gaussian processes and their representations Brownian motion and white noise Some properties of Brownian motion Regularity of Brownian motion Approximation of Brownian motion Brownian motion and stochastic calculus Stochastic chain rule: Ito formula Integration methods in random space Monte Carlo method and its variants Multilevel Monte Carlo method Quasi-Monte Carlo methods Wiener chaos expansion method Stochastic collocation method Smolyak's sparse grid Application to SODEs Bibliographic notes Suggested practice Numerical methods for stochastic differential equations Basic aspects of SODEs Existence and uniqueness of strong solutions Solution methods The integrating factor method Moment equations of solutions Numerical methods for SODEs Derivation of numerical methods based on numerical integration Strong convergence Weak convergence Linear stability Summary of numerical SODEs Basic aspects of SPDEs Functional spaces Solutions in different senses Solutions to SPDEs in explicit form Linear stochastic advection-diffusion-reactionequations Existence and uniqueness Conversion between Ito and Stratonovichformulation Numerical methods for SPDEs Direct semi-discretization methods for parabolicSPDEs Second-order equations Fourth-order equations Wong-Zakai approximation for parabolic SPDEs Preprocessing methods for parabolic SPDEsSplitting methods Integrating factor (exponential integrator) techniques What could go wrong? Examples of stochastic Burgers and Navier-Stokes equations Stability and convergence of existing numericalmethods Weak convergence Pathwise convergence Stability Summary of numerical SPDEs Summary and bibliographic notes Suggested practice Numerical Stochastic Ordinary Differential Equations Numerical schemes for SDEs with time delay using the Wong-Zakai approximation Wong-Zakai approximation for SODEs Wong-Zakai approximation for SDDEs Derivation of numerical schemes A predictor-corrector scheme The midpoint scheme A Milstein-like scheme Linear stability of some schemes Stability region of the forward Euler scheme Stability analysis of the predictor-corrector scheme Stability analysis of the midpoint scheme Numerical results Summary and bibliographic notes Suggested practice Balanced numerical schemes for SDEs with non-Lipschitzcoefficients A motivating example Fundamental theorem On application of Theorem
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