In 1994 and 1998 F. Delbaen and W. Schachermayer published two breakthrough papers in which they proved continuous-time versions of the Fundamental Theorem of Asset Pricing. This is one of the most remarkable achievements in modern Mathematical Finance, which led to intensive investigations in many applications of the arbitrage theory on a mathematically rigorous basis of stochastic calculus.
This book provides the reader with a detailed understanding of all necessary attributes in stochastic calculus that are required for applications of the theory of stochastic integration in Mathematical Finance, in particular, in the arbitrage theory. The exposition follows the traditions of the Strasbourg school. The author provides an overview of the so-called “general theory of stochastic processes”, introducing optional and predictable σ-algebras, predictable stopping times, the decomposition of a stopping time into accessible and totally inaccessible parts. He also studies local martingales and processes of bounded variation, proving the Doob-Meyer decomposition and a decomposition of a local martingale into the sum of a continuous local martingale and a purely discontinuous local martingale and describing the structure of purely discontinuous local martingales. The theory of stochastic integration is then subject to definition with respect to local martingales and semimartingales covering σ-martingale and the Ansel-Stricker lemma, indispensable in the theory of arbitrage, before presenting the most popular applications of the theory of stochastic integration, and defining the stochastic exponential.
1. General Theory of Stochastic Processes.
2. Martingales and Processes with Finite Variation.
3. Stochastic Integrals.
Alexander Gushchin is currently Professor at the Higher School of Economics in Russia, and for the past two years has also been the Leading Scientific Researcher for its International Laboratory of Quantitative Finance (ILQF). The ILQF has been one of the leading institutions in the field of mathematical models of financial markets since it opened in 2013
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