Description
Book SynopsisCompletely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance.
Trade Review“As supplementary reading for a second course or as s comprehensive (!) resource for the general theory of processes aimed at Ph. D. students and scholars, this second edition will stay a valuable resource.” (René L. Schilling, Mathematical Reviews, October, 2016)
“This is a fundamental book in modern stochastic calculus and its applications: rich contents, well structured material, comprehensive coverage of all significant results given with complete proofs and well illustrated by examples, carefully written text. Hence, there are more than enough reasons to strongly recommend the book to a wide audience. Among them, there are good and motivated graduate university students. … Also, the book is an excellent reference book.” (Jordan M. Stoyanov, zbMATH 1338.60001, 2016)
Table of ContentsPart I: Measure Theoretic Probability.- Measure Integral.- Probabilities and Expectation.- Part II: Stochastic Processes.- Filtrations, Stopping Times and Stochastic Processes.- Martingales in Discrete Time.- Martingales in Continuous Time.- The Classification of Stopping Times.- The Progressive, Optional and Predicable -Algebras.- Part III: Stochastic Integration.- Processes of Finite Variation.- The Doob-Meyer Decomposition.- The Structure of Square Integrable Martingales.- Quadratic Variation and Semimartingales.- The Stochastic Integral.- Random Measures.- Part IV: Stochastic Differential Equations.- Ito's Differential Rule.- The Exponential Formula and Girsanov's Theorem.- Lipschitz Stochastic Differential Equations.- Markov Properties of SDEs.- Weak Solutions of SDEs.- Backward Stochastic Differential Equations.- Part V: Applications.- Control of a Single Jump.- Optimal Control of Drifts and Jump Rates.- Filtering. Part VI: Appendices.
