Contents -- Editorial Foreword -- Preface -- 1. Hyperbolic Equations with Random Boundary Conditions Zdzis law Brzeźniak and Szymon Peszat -- Contents -- 1. Introduction -- 1.1. The Wave equation -- 1.2. The Transport equation -- 2. Abstract formulation -- 3. The Wave equation - introduction -- 4. Weak solution to the wave equation -- 5. Mild formulations -- 6. Scales of Hilbert spaces -- 6.1. Application to the boundary value problem -- 7. Equivalence of weak and mild solutions -- 8. The Fundamental Solution -- 9. Applications -- 10. The Transport equation -- References -- 2. Decoherent Information of Quantum Operations Xuelian Cao, Nan Li and Shunlong Luo -- Contents -- 1. Introduction -- 2. Quantum mutual information and purification of mixed states -- 3. Decoherent information -- 4. No-broadcasting in terms of decoherent information -- 5. Continuity of the decoherent information -- 6. Discussion -- Acknowledgement -- References -- 3. Stabilization of Evolution Equations by Noise Tomás Caraballo and Peter E. Kloeden -- Keywords: -- Contents -- 1. Introduction -- 2. Linear PDEs -- 2.1. Persistence of stability and stabilization by Itˆo noise -- 2.2. Destabilization by Itˆo noise -- 3. Linear PDEs without fully commuting noise -- 3.1. Stabilization by simple multiplicative Itˆo noise -- 3.2. Stabilization by Stratonovich noise -- 4. Nonlinear PDEs -- 4.1. Stabilization by Itˆo noise -- 4.2. Stabilization by Stratonovich noise -- 5. Other types of evolution equations and models -- 5.1. Delay di.erential equations -- 5.2. Stabilization of evolution inclusions and PDEs without uniqueness -- 5.3. Stabilization of stationary solutions of a stochastic PDE -- 5.4. Other types of problems -- Acknowledgement -- References -- 4. Stochastic Quantification of Missing Mechanisms in Dynamical Systems Baohua Chen and Jinqiao Duan -- Key Words: -- Contents.

1. Introduction -- 2. Stochastic analysis and stochastic parameterizations -- 3. An example -- Acknowledgements -- References -- 5. Banach Space-Valued Functionals of White Noise Yin Chen and Caishi Wang -- Contents -- 1. Introduction -- 2. Kernel theorems -- 3. Main theorems -- Acknowledgement -- References -- 6. Hurst Index Estimation for Self-Similar Processes with Long-Memory Alexandra Chronopoulou and Frederi G. Viens -- Keywords: -- Contents -- 1. Introduction -- 1.1. Motivation -- 1.2. Mathematical Background -- 2. Most Popular Hurst Parameter Estimators -- 2.1. Heuristic Estimators -- R/S Estimator: -- Correlogram: -- Variogram: -- 2.2. Maximum Likelihood Estimation -- 2.3. Wavelet Estimator -- 3. Multiplication in the Wiener Chaos & Hermite Processes -- 3.1. Basic Tools on Multiple Wiener-Itˆo Integrals -- 3.2. Main Definitions -- 4. Hurst Parameter Estimator Based on Discrete Variations -- 4.1. Estimator Construction -- 4.2. Asymptotic Properties of N -- 5. Comparison & Conclusions -- 5.1. Variations Estimator vs. mle -- 5.2. Variations' vs. Wavelet Estimator -- References -- 7. Modeling Colored Noise by Fractional Brownian Motion Jinqiao Duan, Chujin Li and Xiangjun Wang -- Keywords: -- Contents -- 1. What is noise -- 2. Generalized time derivative and generalized stochastic processes -- 3. White noise -- 4. Colored noises and fractional Brownian motion -- References -- 8. A Sufficient Condition for Non-Explosion for a Class of Stochastic Partial Di.erential Equations Hongbo Fu, Daomin Cao and Jinqiao Duan -- Keywords : -- Contents -- 1. Introduction -- 2. Preliminaries -- 3. Main results -- 4. Examples -- Acknowledgements -- References -- 9. The Influence of Transaction Costs on Optimal Control for an Insurance Company with a New Value Function Lin He, Zongxia Liang and Fei Xing -- Contents -- 1. Introduction.

2. Stochastic Control Model -- 3. Comparisons of the Two Models -- 4. HJB Equation and Verification Theorem -- 5. Construction of Solution to the HJB Equation -- 6. Economic Analysis -- Acknowledgement -- References -- 10. Limit Theorems for p-Variations of Solutions of SDEs Driven by Additive Stable Lévy Noise and Model Selection for Paleo-Climatic Data Claudia Hein, Peter Imkeller and Ilya Pavlyukevich -- Keywords: -- Contents -- 1. Introduction -- 2. Object of study and main results -- 3. Applications to real data -- 4. Functional convergence for V n (L) -- 5. Generalisation to sums of processes -- 5.1. Equivalence for p ≤ 1 -- 5.2. Equivalence for p > α -- 5.3. Equivalence for α ∈ (1, 2), p ∈ (1, α] -- Acknowledgement -- References -- 11. Class II Semi-Subgroups of the Infinite Dimensional Rotation Group and Associated Lie Algebra Takeyuki Hida and Si Si -- Keywords: -- 1. Introduction -- 2. Class II Subgroups of O(E) -- 3. Half whiskers -- 4. Lie algebra -- 5. Concluding remarks -- Acknowledgement -- References -- 12. Stopping Weyl Processes Robin L. Hudson -- Contents -- 1. Introduction -- 2. Characterization of unitary product integrals -- 3. Stop times -- 4. Factorizing product integrals at stop times -- Acknowledgement -- References -- 13. Karhunen-Loéve Expansion for Stochastic Convolution of Cylindrical Fractional Brownian Motions Zongxia Liang -- Keywords: -- Contents -- 1. Introduction -- 2. Preliminaries -- 2.1. Fractional Brownian motion -- 2.2. Hypotheses on operators A and Φ -- 2.3. Stochastic convolution WH A of cylindrical fractional Brownian motion -- 3. Karhunen-Loéve expansion of WH -- 4. Evaluation of conditional exponential moments -- Acknowledgement -- References -- 14. Stein's Method Meets Malliavin Calculus: A Short Survey With New Estimates Ivan Nourdin and Giovanni Peccati -- Keywords: -- Contents -- 1. Introduction.

1.1. Stein's heuristic and method -- 1.2. The role of Malliavin calculus -- 1.3. Beyond the method of moments -- 1.4. An overview of the existing literature -- 2. Preliminaries -- 2.1. Isonormal Gaussian processes -- 2.2. Chaos, hypercontractivity and products -- 2.3. The language of Malliavin calculus -- 3. One-dimensional approximations -- 3.1. Stein's lemma for normal approximations -- 3.2. General bounds on the Kolmogorov distance -- 3.4. Quadratic variation of the fractional Brownian motion, part one -- 3.5. The method of (fourth) moments: explicit estimates via interpolation -- 4. Multidimensional case -- 4.1. Main bounds -- 4.2. Quadratic variation of fractional Brownian motion, continued -- References -- 15. On Stochastic Integrals with Respect to an In.nite Number of Poisson Point Process and Its Applications Guanglin Rang, Qing Li and Sheng You -- Keywords: -- Contents -- 1. Introduction -- 2. Stochastic integral with respect to the infinite number of Poisson point processes -- 3. Martingale representation -- 4. Non-Markovian SDE driven by countably many Poisson point processes -- References -- 16. Lévy White Noise, Elliptic SPDEs and Euclidean Random Fields Jiang-Lun Wu -- Contents -- 1. Introduction -- 2. Lévy white noise -- 3. Lévy white noise and random fields -- 4. Comparison of vector and scalar models -- 5. New formulation of elliptic SPDEs and the lattice approximation -- Acknowledgement -- References -- 17. A Short Presentation of Choquet Integral Jia-An Yan -- Contents -- 1. Introduction -- 2. Integration of Monotone Functions -- 3. Monotone Set Functions, Measurability of Functions -- 4. Comonotonicity of Functions -- 5. The Choquet Integral -- 5.1. Definition and basic properties -- 5.2. Example 1: Distorted probability measures -- 5.3. Example 2: λ-fuzzy measures -- 6. The Subadditivity Theorem.

7. Representing Functionals as Choquet Integrals -- References.