Stochastic calculus for quantitative finance.

Abstract

The arbitrage theory for general models of financial markets in continuous time is based on the heavy use of the theory of martingales and stochastic integration (see the monograph by Delbaen and Schchermayer [DEL 06]). Our book gives an exposition of the foundations of modern theory of stochastic integration (with respect to semimartingales. It follows traditions of the Strasbourg School of Stochastic Processes. In particular, the exposition is inspired by the monograph by Dellacherie [DEL 72]) in Chapter 1 and by the course by Meyer [MEY 76] in Chapters 2 and 3. In Chapter 1, the so-called general theory of stochastic processes is developed. The second chapter is devoted to detailed study of local martingales and processes with finite variation. The theory of stochastic integration with respect to semimartingales is a subject of Chapter 3. We do not consider vector stochastic integrals, for which we refer to Shiryaev and Cherny [SHI 02]. The last section is devoted to σ-martingales and the Ansel–Stricker theorem. Some results are given without proofs. These include the section theorem, classical Doob’s theorems on martingales, the Burkholder–Davis–Gundy inequality and Itô’s formula. Our method of presentation may be considered as old-fashioned, compared to, for example, the monograph by Protter [PRO 05], which begins with an introduction of the notion of a semimartingale; in our book, semimartingales appear only in the final chapter. However, the author’s experience based on the graduate courses taught at the Department of Mechanics and Mathematics of Moscow State University, indicates that our approach has some advantages. The text is intended for a reader with a knowledge of measure-theoretic probability and discrete-time martingales. Some information on less standard topics (theorems on monotone classes, uniform integrability, conditional expectation for nonintegrable random variables and functions of bounded variation) can be found in the Appendix. The basic idea, which the author pursued when writing this book, was to provide an affordable and detailed presentation of the foundations of the theory of stochastic integration, which the reader needs to know before reading more advanced literature on the subject, such as Jacod [JAC 79], Jacod and Shiryaev [JAC 03], Liptser and Shiryayev [LIP 89], or a literature dealing with applications, such as Delbaen and Schchermayer [DEL 06]. The text is accompanied by more than a hundred exercises. Almost all of them are simple or are supplied with hints. Many exercises extend the text and are used later. The work on this book was partially supported by the International Laboratory of Quantitative Finance, National Research University Higher School of Economics and Russian Federation Government (grant no. 14.A12.31.0007). I wish to express my sincere thanks to Tatiana Belkina for a significant and invaluable assistance in preparing the manuscript.