Introduction to Stochastic Calculus

Description

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. 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