- Discusses quadratic variation of a square integrable martingale, pathwise formulae for the stochastic integral, Emery topology, and sigma-martingales
- Uses the technique of random time change to study the solution of a stochastic differential equation (SDE) driven by continuous semi-martingales
- Studies the predictable increasing process to introduce predictable stopping times and to prove the Doob–Meyer decomposition theorem
- Gives an extensive treatment of representation of martingales as stochastic integrals
- Is useful for a two-semester graduate-level course on measure-theoretic probability
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- About this Textbook
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This book sheds new light on stochastic calculus, the branch of mathematics that is most widely applied in financial engineering and mathematical finance. The first book to introduce pathwise formulae for the stochastic integral, it provides a simple but rigorous treatment of the subject, including a range of advanced topics. The book discusses in-depth topics such as quadratic variation, Ito formula, and Emery topology. The authors briefly addresses continuous semi-martingales to obtain growth estimates and study solution of a stochastic differential equation (SDE) by using the technique of random time change. Later, by using Metivier–Pellaumail inequality, the solutions to SDEs driven by general semi-martingales are discussed. The connection of the theory with mathematical finance is briefly discussed and the book has extensive treatment on the representation of martingales as stochastic integrals and a second fundamental theorem of asset pricing. Intended for undergraduate- and beginning graduate-level students in the engineering and mathematics disciplines, the book is also an excellent reference resource for applied mathematicians and statisticians looking for a review of the topic.
- About the authors
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Rajeeva Laxman Karandikar has been professor and director of Chennai Mathematical Institute, Tamil Nadu, India, since 2010. An Indian mathematician, statistician and psephologist, Prof. Karandikar is a fellow of the Indian Academy of Sciences, Bengaluru, India, and the Indian National Science Academy, New Delhi, India. He received his MStat and PhD from the Indian Statistical Institute, Kolkata, India, in 1978 and 1981, respectively. He spent two years as a visiting professor at the University of North Carolina, Chapel Hill, USA, and worked with Prof. Gopinath Kallianpur. He returned to the Indian Statistical Institute, New Delhi, India, in 1984. In 2006, he moved to Cranes Software International Limited, where he was executive vice president for analytics until 2010. His research interests include stochastic calculus, filtering theory, option pricing theory, psephology in the context of Indian elections and cryptography, among others.
B.V. Rao is an adjunct professor at Chennai Mathematical Institute, Tamil Nadu, India. He received his MSc degree in Statistics from Osmania University, Hyderabad, India, in 1965 and the doctoral degree from the Indian Statistical Institute, Kolkata, India, in 1970. His research interests include descriptive set theory, analysis, probability theory and stochastic calculus. He was a professor and later a distinguished scientist at the Indian Statistical Institute, Kolkata. Generations of Indian probabilists have benefitted from his teaching, where he taught from 1973 till 2009.
- Table of contents (13 chapters)
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Discrete Parameter Martingales
Pages 1-33
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Continuous-Time Processes
Pages 35-63
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The Ito’s Integral
Pages 65-87
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Stochastic Integration
Pages 89-160
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Semimartingales
Pages 161-213
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Table of contents (13 chapters)
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Bibliographic Information
- Bibliographic Information
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- Book Title
- Introduction to Stochastic Calculus
- Authors
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- R. L. Karandikar
- B. V. Rao
- Series Title
- Indian Statistical Institute Series
- Copyright
- 2018
- Publisher
- Springer Singapore
- Copyright Holder
- Springer Nature Singapore Pte Ltd.
- eBook ISBN
- 978-981-10-8318-1
- DOI
- 10.1007/978-981-10-8318-1
- Hardcover ISBN
- 978-981-10-8317-4
- Softcover ISBN
- 978-981-13-4121-2
- Series ISSN
- 2523-3114
- Edition Number
- 1
- Number of Pages
- XIII, 441
- Topics
