Chapter 1. Introduction 1.1. Notation Chapter 2. Strong stability analysis for solutions of SDEs 2.1. Setting 2.2. Exponential integrability bounds for solutions of SDEs 2.3. An identity for Lyapunov-type functions 2.4. Two solution approach Chapter 3. Strong completeness of SDEs 3.1. Theorems of Yamada-Watanabe and of Kolmogorov-Chentsov type 3.2. Proofs of the strong completeness results 3.3. Strong completeness for SDEs with additive noise Chapter 4. Examples of SODEs 4.1. Setting 4.2. Stochastic van der Pol oscillator 4.3. Stochastic Duffing-van der Pol oscillator 4.4. Stochastic Lorenz equation with additive noise 4.5. Langevin dynamics 4.6. Brownian dynamics (Over-damped Langevin dynamics) 4.7. Stochastic SIR model 4.8. Experimental psychology model 4.9. Stochastic Brusselator in the well-stirred case 4.10. Stochastic volatility processes and interest rate models (CIR, Ait-Sahalia, 3/2-model, CEV) 4.11. Wright-Fisher diffusion Chapter 5. Examples of SPDEs 5.1. Setting 5.2. Stochastic Burgers equation with a globally bounded diffusion coefficient and trace class noise 5.3. Cahn-Hilliard Cook equation with trace class noise 5.4. Non-linear wave equation Acknowledgments Bibliography

Let X be a (quasi-)Banach space. Let d = (dn) n≥1 be an X-valued sequence of random variables adapted to a filtration (Fn) n≥1 on a probability space (Ω, A, P), define F∞ := σ(Fn : n ≥ 1) and let e = (en) n≥1 be a F∞-conditionally independent sequence on (Ω, A, P) such that L(dn | F n−1 ) = L(en | F∞) for all n ≥ 1 (F 0 = {Ω, ∅}). If there exists a p ∈ (0, ∞) and a constant Dp independent of d and e such that one has, for all n ≥ 1,

In this article we prove pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problemHere A is the generator of an analytic C 0 -semigroup on a umd Banach space X, W H is a cylindrical Brownian motion in a Hilbert space H, and the functions F : [0, T ] × X → X θ F and G : [0, T ] × X → L (H, X θ G ) satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise. γ ([0,T ];X) 1 p ≤ Cn −δ (1 + x 0 L p (Ω;Xη ) ).

Abstract. Let E be a type 2 umd Banach space, H a Hilbert space and let p ∈ [1, ∞). Consider the following stochastic delay equation in E:whereis given by a Riemann-Stieltjes integral, and B :We prove that a solution to (SDE) is equivalent to a solution to the corresponding stochastic Cauchy problem, and use this to prove the existence, uniqueness and continuity of a solution to (SDE).