Itô's Stochastic Calculus and Probability Theory

ISBN-13: 9784431685340 / Angielski / Miękka / 2012 / 422 str.

Nobuyuki Ikeda; Sinzo Watanabe; Masatoshi Fukushima

Opis

Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics.
This volume contains 27 papers written by world-renowned probability theorists. Their subjects vary widely and they present new results and ideas in the fields where stochastic analysis plays an important role. Also included are several expository articles by well-known experts surveying recent developments. Not only mathematicians but also physicists, biologists, economists and researchers in other fields who are interested in the effectiveness of stochastic theory will find valuable suggestions for their research. In addition, students who are beginning their study and research in stochastic analysis and related fields will find instructive and useful guidance here.
This volume is dedicated to Professor Ito on the occasion of his eightieth birthday as a token of deep appreciation for his great achievements and contributions. An introduction to and commentary on the scientific works of Professor Ito are also included.

Spis treści

Lévy measure of superprocesses; Absorption processes.- A class of integration by parts formulae in stochastic analysis I.- Smooth measures and continuous additive functionals of right Markov processes.- On the decomposition of additive functionals of reflecting Brownian motions.- Equilibrium fluctuations for lattice gas.- Hall’s transform and the Segal-Bargmann map.- Lagrangian for pinned diffusion process.- Short time asymptotics and an approximation for the heat kernel of a singular diffusion.- Van Vleck-Pauli formula for Wiener integrals and Jacobi fields.- Some recent developments in nonlinear filtering theory.- Detecting a single defect in a scenery by observing the scenery along a random walk path.- Analytic approach to Yor’s formula of exponential additive functionals of Brownian motion.- Stochastic differential equations with jumps and stochastic flows of diffeomorphisms.- A remark on American securities.- Calculus for multiplicative functionals, Itô’s formula and differential equations.- A Martin boundary connected with the ?-volume limit of the focussing cubic Schrödinger equation.- Diffusion processes on an open time interval and their time reversal.- On sensitive control and differential games in infinite dimensional space.- Decomposition at the maximum for excursions and bridges of one-dimensional diffusions.- Interacting diffusion systems over Zd.- A Kähler metric on a based loop group and a covariant differentiation.- Burgers system driven by a periodic stochastic flow.- An estimate on the Hessian of the heat kernel.- Environment-wise central limit theorem for a diffusion in a Brownian environment with large drift.- The complex story of simple exclusion.- Lévy’s stochastic area formula and Brownian motion on compact Lie groups.- Principal values of Brownian local times and their related topics.