Notes on stochastic integration theory with respect to c\`adl\`ag semimartingales and a brief introduction to L\'evy processes

Abstract

The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in applying these concepts, particularly in the study of stochastic (functional) differential equations driven by L\'evy processes. These notes are adapted from the preliminary chapter of the author's master's thesis (with only minor changes) and are intended to introduce newcomers to the essentials of c\`adl\`ag semimartingale theory while also discussing the advantages, limitations, and subtleties as compared to stochastic integration in the continuous setting.