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基本説明
The emphasis of this book is on classes of Brownian functionals like: - Gaussian subspaces of the Gaussian space of Brownian motion; - Brownian quadratic functionals; - Brownian local times, - Exponential functionals of Brownian motion with drift ; and more.
Full Description
Stochastic calculus and excursion theory are very efficient tools to obtain either exact or asymptotic results about Brownian motion and related processes. The emphasis of this book is on special classes of such Brownian functionals as:
- Gaussian subspaces of the Gaussian space of Brownian motion;
- Brownian quadratic funtionals;
- Brownian local times,
- Exponential functionals of Brownian motion with drift;
- Winding number of one or several Brownian motions around one or several points or a straight line, or curves;
- Time spent by Brownian motion below a multiple of its one-sided supremum.
Besides its obvious audience of students and lecturers the book also addresses the interests of researchers from core probability theory out to applied fields such as polymer physics and mathematical finance.
Contents
The Gaussian space of BM.- The laws of some quadratic functionals of BM.- Squares of Bessel processes and Ray-Knight theorems for Brownian local times.- An explanation and some extensions of the Ciesielski-Taylor identities.- On the winding number of planar BM.- On some exponential functionals of Brownian motion and the problem of Asian options.- Some asymptotic laws for multidimensional BM.- Some extensions of Paul Lévy's arc sine law for BM.- Further results about reflecting Brownian motion perturbed by its local time at 0.- On principal values of Brownian and Bessel local times.- Probabilistic representations of the Riemann zeta function and some generalisations related to Bessel processes.
