zbMATH — the first resource for mathematics

The mathematics of arbitrage. (English) Zbl 1106.91031
The book consists of two parts. Part I contains the “guided tour” through the mathematical theory of finances which is divided into eight chapters. Chapters 1–4 give expositions of basic topics of mathematical finance and are kept at an elementary technical level. From Chapter 5 on, the level of technical sophistication increases rather steeply. Chapter 5 starts with D. Krep’s version of the Fundamental Theorem of Asset Pricing involving the notion of “No Free Lunch”. In the next chapters several different proofs of the Dalang-Morton-Willinger theorem are included, in particular, the proof based on the notion of “measurably parameterised subsequences”. Also, a quick overview of stochastic integration is presented.
Part II consists of updated versions of seven papers of the authors devoted to financial mathematics with a number of corrections. Among them “A general version of the Fundamental Theorem of Asset Pricing’, “The existence of absolutely continuous local martingale measures” and some others that became classics in mathematical finance.

91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91Gxx Actuarial science and mathematical finance
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)