A Crash Course in Stochastic Calculus with Applications to Mathematical Finance

Introduction The purpose of this paper is to summarize a number of results from stochastic calculus that are of fundamental importance in advanced mathematical nance The emphasis will be on the concepts from probability theory stochastic pro cesses and stochastic integration theory Throughout the paper we will hint at applications of the presented results to problems in mathematical nance that are treated in Section A variety of good textbooks on the subject is available each with its own avour We mention Rogers and Williams Revuz and Yor Chung and Williams ksendal Protter For the prepara tion of this paper I heavily leaned on the book by Karatzas and Shreve which was my main source of inspiration Readers are supposed to be familiar with some measure theory and the measure theoretic foundation of probability In these notes proofs are usually not given although sometimes key steps of a proof are indicated to give the reader an idea of the avour of the methods that are used