Introduction to Stochastic Calculus

✍ Scribed by Rajeeva L. Karandikar, B. V. Rao

Publisher: Springer Singapore
Year: 2018
Tongue: English
Leaves: 446
Series: Indian Statistical Institute Series
Edition: 1st ed.
Category: Library

No coin nor oath required. For personal study only.

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-xiii
Discrete Parameter Martingales (Rajeeva L. Karandikar, B. V. Rao)....Pages 1-33
Continuous-Time Processes (Rajeeva L. Karandikar, B. V. Rao)....Pages 35-63
The Ito’s Integral (Rajeeva L. Karandikar, B. V. Rao)....Pages 65-87
Stochastic Integration (Rajeeva L. Karandikar, B. V. Rao)....Pages 89-160
Semimartingales (Rajeeva L. Karandikar, B. V. Rao)....Pages 161-213
Pathwise Formula for the Stochastic Integral (Rajeeva L. Karandikar, B. V. Rao)....Pages 215-220
Continuous Semimartingales (Rajeeva L. Karandikar, B. V. Rao)....Pages 221-249
Predictable Increasing Processes (Rajeeva L. Karandikar, B. V. Rao)....Pages 251-302
The Davis Inequality (Rajeeva L. Karandikar, B. V. Rao)....Pages 303-320
Integral Representation of Martingales (Rajeeva L. Karandikar, B. V. Rao)....Pages 321-360
Dominating Process of a Semimartingale (Rajeeva L. Karandikar, B. V. Rao)....Pages 361-381
SDE Driven by r.c.l.l. Semimartingales (Rajeeva L. Karandikar, B. V. Rao)....Pages 383-410
Girsanov Theorem (Rajeeva L. Karandikar, B. V. Rao)....Pages 411-434
Back Matter ....Pages 435-441

✦ Subjects

Statistics; Statistical Theory and Methods; Probability Theory and Stochastic Processes

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