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.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 the world were invited to present the newest developments within the exciting and fast growing field of stochastic analysis. The present volume combines both papers from the invited speakers and contributions by the presenting lecturers.A special feature is the Memoirs that Kiyoshi Ito wrote for this occasion. These are valuable pages for both young and established researchers in the field.

Memoirs of My Research on Stochastic Analysis by K.Ito.- Ito calculus and quantum white noise calculus by L. Accardi, A. Boukas.- Homogenization of diffusions on the lattice Zd with periodic drift coefficients, applying a logarithmic Sobolev inequality or a weak Poincare inequality by S. Albeverio, M. S. Bernabei, M. Röckner, M. W. Yoshida.- Theory and applications of infinite dimensional oscillatory integrals by S. Albeverio, S. Mazzucchi.- Ambit processes with applications to turbulence and tumour growth by O. E. Barndorff-Nielsen, J. Schmiegel.- A stochastic control approach to a robust utility maximization problem by G. Bordigoni, A. Matoussi, M. Schweizer.- Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions by Zhen-Qing Chen, M. Fukushima, J. Ying.- Hedging with options in models with jumps by R. Cont, P. Tankov, E. Voltchkova.- Power variation analysis of some integral long-memory processes by J. M. Corcuera.- Kolmogorov equations for stochastic PDE¿s with multiplicative noise by G. Da Prato.- Stochastic Integrals and Adjoint Derivatives by G. Di Nunno, Yu.A. Rozanov.- An application of probability to nonlinear analysis by E.B. Dynkin.- The space of stochastic differential equations by K.D. Elworthy.- Extremes of supOU processes by V. Fasen, C. Klüppelberg.- Gaussian bridges by D. Gasbarra, T. Sottinen, E. Valkeila.- Some of the recent topics on stochastic analysis by T. Hida.- Differential equations driven by Hölder continuous functions of order greater than ¿ by Y. Hu, D. Nualart.- On asymptotics of Banach space-valued Ito functionals of Brownian rough paths by Y. Inahama, H. Kawabi.- Continuous-Time Markowitz¿s Problems in an Incomplete Market, with No-Shorting Portfolios by H. Jin, Xun Yu Zhou.- Quantum and Classical Conserved Quantities: Martingales, Conservation Laws and Constants of Motion by T. Kolsrud.- Different Lattice Approximations for Hoegh-Krohn¿s quantum field model by S. Liang.- Ito Atlas, itsapplication to mathematical finance and to exponentiation of infinite dimensional Lie algebras by P. Malliavin.- The Invariant Distribution of a Diffusion: Some New Aspects by H.P. McKean.- Formation of singularities in Madelung fluid: a nonconventional application of Ito calculus to foundations of Quantum Mechanics by L.M. Morato.- G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito Type by S. Peng.- Perpetual integral functionals of diffusions and their numerical computations by P. Salminen, O. Wallin.- Chaos expansions and Malliavin calculus for Levy processes by J. Lluis Sole, Frederic Utzet, Josep Vives.- Study of Simple but Challenging Diffusion Equation by D.W. Stroock.- Ito Calculus and Malliavin Calculus by S. Watanabe.- The Malliavin calculus for processes with conditionally independent increments by A.L. Yablonski