Упражнения и решения с ключевыми результатами по локальному времени, броуновским экскурсиям и полумартингалам: стохастические процессы, броуновское движение, исчисление Ито, теория экскурсий

This book offers a thorough introduction to the local times of continuous semimartingales and the theory of Brownian excursions, through a progressive approach based on detailed exercises. The first part is devoted to the fundamental tools of stochastic calculus: Itô’s formula, the Tanaka–Meyer formula, occupation density, and essential properties of local times. These concepts allow for a fine analysis of stochastic process paths in terms of the time spent near a given level. The second part develops the theory of Brownian excursions, which provides a remarkable decomposition of paths into independent segments. This approach leads to deep results in probability, including Lévy’s, Pitman’s, and Ray–Knight theorems, as well as applications such as the Feynman–Kac formula. The book emphasizes the interplay between mathematical rigor, probabilistic intuition, and computational techniques. It is primarily aimed at Master’s students in probability, as well as any reader seeking to acquire a solid command of modern tools in stochastic analysis.