Brownian local time

Abstract

CONTENTS Introduction Chapter I. Distributions of functionals of Brownian local time § 1. Brownian local time § 2. Distributions of Brownian local time § 3. The Markov property § 4. Distributions of integral functionals of Brownian local time and functionals of supremum type § 5. Distributions of functionals of occupation time type § 6. Distribution of the supremum of the increments of Brownian local time Chapter II. Properties of trajectories of Brownian local time § 1. Continuity § 2. Law of the iterated logarithm § 3. Brownian local time - a quasi-martingale Chapter III. Convergence of stochastic processes to Brownian local time § 1. Brownian local time as a limit process § 2. Hausdorff measure of the level sets of Brownian motion § 3. On the character of the convergence to Brownian local time § 4. Limit theorems for the occupation times of Brownian motion § 5. On estimation of Brownian local time from observations of Brownian motion at discrete instants Chapter IV. Invariance principle for local times § 1. Weak invariance principle for local times § 2. Strong invariance principle for local times § 3. Limit behaviour of the differences between Brownian local time and processes converging to it § 4. Law of the iterated logarithm for local time of a random walk Chapter V. Applications of Brownian local time § 1. The generalized Itô formula § 2. Non-negative continuous homogeneous additive functionals of Brownian motion References