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© 2018

Introduction to Stochastic Calculus

Textbook
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Part of the Indian Statistical Institute Series book series (INSIS)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Rajeeva L. Karandikar, B. V. Rao
    Pages 1-33
  3. Rajeeva L. Karandikar, B. V. Rao
    Pages 35-63
  4. Rajeeva L. Karandikar, B. V. Rao
    Pages 65-87
  5. Rajeeva L. Karandikar, B. V. Rao
    Pages 89-160
  6. Rajeeva L. Karandikar, B. V. Rao
    Pages 161-213
  7. Rajeeva L. Karandikar, B. V. Rao
    Pages 215-220
  8. Rajeeva L. Karandikar, B. V. Rao
    Pages 221-249
  9. Rajeeva L. Karandikar, B. V. Rao
    Pages 251-302
  10. Rajeeva L. Karandikar, B. V. Rao
    Pages 303-320
  11. Rajeeva L. Karandikar, B. V. Rao
    Pages 321-360
  12. Rajeeva L. Karandikar, B. V. Rao
    Pages 361-381
  13. Rajeeva L. Karandikar, B. V. Rao
    Pages 383-410
  14. Rajeeva L. Karandikar, B. V. Rao
    Pages 411-434
  15. Back Matter
    Pages 435-441

About this book

Introduction

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 address 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.

Keywords

Stochastic Calculus Martingale Convergence Theorem Continuous Time Process The Ito Integral Stochastic Integration Semimartingales

Authors and affiliations

  1. 1.Chennai Mathematical InstituteSiruseriIndia
  2. 2.Chennai Mathematical InstituteSiruseriIndia

About the authors

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.

Bibliographic information

Reviews

“The style is compact and clear. The presentation is well complemented by a large number of useful remarks and exercises. Graduate students attending university courses in modern probability theory and its applications can benefit a lot from working with this book. There are good reasons to expect that the book will be met positively by students, university teachers and young researchers.” (Jordan M. Stoyanov, zbMATH 1434.60003, 2020)