Skip to main content
SpringerLink
Log in
Book cover

Introduction to Stochastic Calculus

  • Textbook
  • © 2018

Overview

Authors:
  1. Rajeeva L. Karandikar

    1. Chennai Mathematical Institute, Siruseri, India

    View author publications

    You can also search for this author in PubMed   Google Scholar

  2. B. V. Rao

    1. Chennai Mathematical Institute, Siruseri, India

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • 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

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

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (13 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Discrete Parameter Martingales

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 1-33

  3. Continuous-Time Processes

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 35-63

  4. The Ito’s Integral

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 65-87

  5. Stochastic Integration

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 89-160

  6. Semimartingales

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 161-213

  7. Pathwise Formula for the Stochastic Integral

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 215-220

  8. Continuous Semimartingales

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 221-249

  9. Predictable Increasing Processes

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 251-302

  10. The Davis Inequality

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 303-320

  11. Integral Representation of Martingales

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 321-360

  12. Dominating Process of a Semimartingale

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 361-381

  13. SDE Driven by r.c.l.l. Semimartingales

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 383-410

  14. Girsanov Theorem

    • Rajeeva L. Karandikar, B. V. Rao
    Pages 411-434

  15. Back Matter

    Pages 435-441
    Download chapter PDF
Back to top

Keywords

About this book

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.

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)

Authors and Affiliations

  • Chennai Mathematical Institute, Siruseri, India

    Rajeeva L. Karandikar, B. V. Rao

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

Publish with us

Policies and ethics

Back to top

Search

Navigation