Optional Processes: Stochastic Calculus and Applications, 1st Edition (Hardback) book cover

Optional Processes

Stochastic Calculus and Applications, 1st Edition

By Mohamed Abdelghani, Alexander Melnikov

Chapman and Hall/CRC

378 pages

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Hardback: 9781138337268
pub: 2020-06-12
Available for pre-order. Item will ship after 12th June 2020

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It is well-known that modern stochastic calculus has been exhaustively developed under standard conditions. Despite such a well-developed theory, there is evidence to suggest that these very convenient technical conditions cannot necessarily be fulfilled in real-world applications.

Optional Processes: Stochastic Calculus and Applications is a book that seeks to delve into the existing theory, new developments and applications of optional processes on "unusual" probability spaces. The development of stochastic calculus of optional processes marks the beginning of a new and more general form of stochastic analysis.

This book aims to provide an accessible, comprehensive and up-to-date exposition of optional processes and their numerous properties. Furthermore, the book presents not only current theory of optional processes, but it also contains a spectrum of applications to stochastic differential equations, filtering theory and mathematical finance.


  • Suitable for graduate students and researchers in mathematical finance, actuarial science, applied mathematics and related areas

  • Compiles almost all essential results on the calculus of optional processes in unusual probability spaces
  • Contains many advanced analytical results for stochastic differential equations and statistics pertaining to the calculus of optional processes
  • Develops new methods in finance based on optional processes such as a new portfolio theory, defaultable claim pricing mechanism etc.

Table of Contents

1. Spaces, Laws and Limits. 2. Stochastic Processes. 3. Martingales. 4. Strong Supermartingales. 5. Optional Martingales. 6. Optional Supermartingales Decomposition. 7. Calculus of Optional Semimartingales. 8. Optional Stochastic Equations. 9. Optional Financial Markets. 10. Defaultable Markets on Unusual Space. 11. Filtering of Optional Semimartingales. Bibliography. Index.

About the Authors

Mohamed Abdelghani completed his PhD in Mathematical Finance from the University of Alberta. He is currently working as a V.P. in quantitative finance and machine learning at Morgan Stanley, New York, USA.

Alexander Melnikov is a Professor in Mathematical Finance at the University of Alberta, Edmonton, Canada. His research interests belong to the area of contemporary stochastic analysis and its numerous applications in Mathematical Finance, Statistics and Actuarial Science. He has written six books as well as over one hundred research papers in leading academic journals.

About the Series

Chapman and Hall/CRC Financial Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General