The Abel Symposium 2005 was organized as a tribute to the work of Kiyosi Ito on the occasion of his 90th birthday. Distinguished researchers from all over presented the newest developments within the exciting and fast growing field of stochastic analysis. This volume combines both papers from the invited speakers and contributions by the presenting lecturers. In addition, it includes the Memoirs that Kiyoshi Ito wrote for this occasion.

"Kiyosi Ito, the founder of stochastic calculus, is one of the few central figures of the twentieth century mathematics who reshaped the mathematical world. Today stochastic calculus is a central research field with applications in several other mathematical disciplines, for example physics, engineering, biology, economics and finance.

Memoirs of My Research on Stochastic Analysis.- Itô Calculus and Quantum White Noise Calculus.- Homogenization of Diffusions on the Lattice Zd with Periodic Drift Coefficients, Applying a Logarithmic Sobolev Inequality or a Weak Poincaré Inequality.- Theory and Applications of Infinite Dimensional Oscillatory Integrals.- Ambit Processes; with Applications to Turbulence and Tumour Growth.- A Stochastic Control Approach to a Robust Utility Maximization Problem.- Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions.- Hedging with Options in Models with Jumps.- Power Variation Analysis of Some Integral Long-Memory Processes.- Kolmogorov Equations for Stochastic PDE's with Multiplicative Noise.- Stochastic Integrals and Adjoint Derivatives.- An Application of Probability to Nonlinear Analysis.- The Space of Stochastic Differential Equations.- Extremes of supOU Processes.- Gaussian Bridges.- Some of the Recent Topics on Stochastic Analysis.- Differential Equations Driven by Hölder Continuous Functions of Order Greater than 1/2.- On Asymptotics of Banach Space-valued Itô Functionals of Brownian Rough Paths.- Continuous-Time Markowitz's Problems in an Incomplete Market, with No-Shorting Portfolios.- Quantum and Classical Conserved Quantities: Martingales, Conservation Laws and Constants of Motion.- Different Lattice Approximations for Hôegh-Krohn's Quantum Field Model.- Itô Atlas, its Application to Mathematical Finance and to Exponentiation of Infinite Dimensional Lie Algebras.- The Invariant Distribution of a Diffusion: Some New Aspects.- Formation of Singularities in Madelung Fluid: A Nonconventional Application of Itô Calculus to Foundations of Quantum Mechanics.- G-Expectation, G-Brownian Motion and Related Stochastic Calculus of ItôType.- Perpetual Integral Functionals of Diffusions and their Numerical Computations.- Chaos Expansions and Malliavin Calculus for Lévy Processes.- Study of Simple but Challenging Diffusion Equation.- Itô Calculus and Malliavin Calculus.- The Malliavin Calculus for Processes with Conditionally Independent Increments.