A mild Ito formula for SPDEs

Abstract

This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical approximation processes.

39 pages, 0 figures

References in corpus (13)

• Stochastic integration in UMD Banach spaces
• A random map implementation of implicit filters
• Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise
• An exponential integrator scheme for time discretization of nonlinear stochastic wave equation
• Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise
• Efficient simulation of nonlinear parabolic SPDEs with additive noise
• Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
• Taylor expansions of solutions of stochastic partial differential equations with additive noise
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• Cubature on Wiener space in infinite dimension
• Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs
• Pathwise Taylor Expansions for Itô Random Fields
• Maximal $L^p$-regularity for stochastic evolution equations

Cited by in corpus (25)

• Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise
• Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations
• Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks
• Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes
• Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients
• An exponential Wagner-Platen type scheme for SPDEs
• Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations
• Application of the Method of Approximation of Iterated Ito Stochastic Integrals Based on Generalized Multiple Fourier Series to the High-Order Strong Numerical Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise
• Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on PYTHON and MATLAB
• Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise
• Application of Multiple Fourier-Legendre Series to Implementation of Strong Exponential Milstein and Wagner-Platen Methods for Non-Commutative Semilinear Stochastic Partial Differential Equations
• Exponential moments for numerical approximations of stochastic partial differential equations
• Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
• Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
• Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient
• Four New Forms of the Taylor-Ito and Taylor-Stratonovich Expansions and its Application to the High-Order Strong Numerical Methods for Ito Stochastic Differential Equations
• Stochastic Lotka-Volterra Competitive Reaction-Diffusion Systems Perturbed by Space-Time White Noise: Modeling and Analysis
• Quadratic covariations for the solution to a stochastic heat equation
• Approximation of SPDEs with Holder Continuous Drifts
• Approximation of SPDE covariance operators by finite elements: A semigroup approach
• Infinite dimensional weak Dirichlet processes and convolution type processes
• Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions