Chapter 1 Introduction
1．1 Hilbert Space and Hilbert Space-Valued Wiener Processes
1．2 The Hilbert Space-Valued Stochastic ItS-Integral
Chapter 2 Error estimates of finite element method for semi-linear stochastic strongly damped wave equation
2．1 The stochastic strongly damped wave equation
2．2 Error estimates for the finite element semi-discret ization and full-discretization of the deterministic linear problem
2．2．1 Error estimates of semi-discrete scheme
2．2．2 Error estimates of the full-discrete scheme
2．3 Finite element method for the stochastic problem
2．3．1 Spatial semi-discretization
2．3．2 Full-discretization
Chapter 3 An accelerated exponential time integrator for semi linear stochastic strongly damped wave equation with additive noise
3．1 Preliminaries and abstract framework
3．2 The proposed scheme and main result
3．3 Proof of the main result
Chapter 4 Optimal error estimates of Galerkin finite element meth-ods for stochastic Allen-Cahn equation with additive noise
4．1 The stochastic Allen-Cahn equation
4．1．1 Abstract framework and main assumptions
4．1．2 Regularity results of the model
4．2 Error estimates of the spatial semi-discretization
4．3 Error estimates of the spatio-temporal fully discretization
Chapter 5 Time-stepping error bound for a stochastic parabolic Volterra equation disturbed by fractional Brownian motions
5．1 Preliminaries
5．1．1 Main assumptions
5．1．2 The linear deterministic problem
5．1．3 The continuous stochastic problem
5．1．4 Numerical methods for stochastic problem
5．2 Error estimates for the deterministic problem
5．3 Convergence rates for the semi-discretization of SPVEs
Bibliography
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