Stochastic Processes

This chapter provides a comprehensive overview of stochastic processes, an essential concept in probability theory and its applications. Beginning with an introduction to both discrete and continuous stochastic processes, we explore fundamental aspects of process similarity, filtrations, and the crucial concept of stopping times. The exposition then transitions to a detailed discussion on martingales, a class of stochastic processes with profound implications in financial mathematics, signal processing, and beyond. We delve into discrete martingales before extending our analysis to continuous martingales, where we introduce the core inequalities and the pivotal Doob-Meyer decomposition.