Stochastic Calculus and Applications

✍ Scribed by Samuel N. Cohen, Robert J. Elliott (auth.)

• Publisher: Birkhäuser Basel
• Year: 2015
• Tongue: English
• Leaves: 673
• Series: Probability and Its Applications
• Edition: 2
• Category: Library

✦ Synopsis

Completely 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. Building upon the original release of this title, this text will be of great interest to research mathematicians and graduate students working in those fields, as well as quants in the finance industry.

New features of this edition include:

End of chapter exercises; New chapters on basic measure theory and Backward SDEs; Reworked proofs, examples and explanatory material; Increased focus on motivating the mathematics; Extensive topical index.

"Such a self-contained and complete exposition of stochastic calculus and applications fills an existing gap in the literature. The book can be recommended for first-year graduate studies. It will be useful for all who intend to work with stochastic calculus as well as with its applications."–Zentralblatt (from review of the First Edition)

✦ Table of Contents

Front Matter....Pages i-xxiii
Front Matter....Pages 1-1
Measure and Integral....Pages 3-47
Probabilities and Expectation....Pages 49-69
Front Matter....Pages 71-71
Filtrations, Stopping Times and Stochastic Processes....Pages 73-87
Martingales in Discrete Time....Pages 89-107
Martingales in Continuous Time....Pages 109-137
The Classification of Stopping Times....Pages 139-151
The Progressive, Optional and Predictable σ-Algebras....Pages 153-171
Front Matter....Pages 173-173
Processes of Finite Variation....Pages 175-197
The Doob–Meyer Decomposition....Pages 199-210
The Structure of Square Integrable Martingales....Pages 211-232
Quadratic Variation and Semimartingales....Pages 233-258
The Stochastic Integral....Pages 259-292
Random Measures....Pages 293-334
Front Matter....Pages 335-335
Itô’s Differential Rule....Pages 337-365
The Exponential Formula and Girsanov’s Theorem....Pages 367-396
Lipschitz Stochastic Differential Equations....Pages 397-426
Markov Properties of SDEs....Pages 427-450
Weak Solutions of SDEs....Pages 451-465
Backward Stochastic Differential Equations....Pages 467-493
Front Matter....Pages 495-495
Control of a Single Jump....Pages 497-516
Front Matter....Pages 495-495
Optimal Control of Drifts and Jump Rates....Pages 517-534
Filtering....Pages 535-566
Back Matter....Pages 567-666

✦ Subjects

Probability Theory and Stochastic Processes; Partial Differential Equations; Electrical Engineering; Computational Mathematics and Numerical Analysis; Quantitative Finance