Introduction to Stochastic Calculus

Defines quadratic variation of a square integrable martingale
Demonstrates pathwise formulae for the stochastic integral
Uses the technique of random time change to study the solution of a stochastic differential equation
Studies the predictable increasing process to introduce predictable stopping times and prove the Doob Meyer decomposition theorem
Is useful for a two-semester graduate level course on measure theory and probability

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 DoobMeyer 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

Autorentext

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.

Inhalt
Discrete Parameter Martingales.- Continuous Time Processes.- The Ito Integral.- Stochastic Integration.- Semimartingales.- Pathwise Formula for the Stochastic Integral.- Continuous Semimartingales.- Predictable Increasing Processes.- The Davis Inequality.- Integral Representation of Martingales.- Dominating Process of a Semimartingale.- SDE driven by r.c.l.l. Semimartingales.- Girsanov Theorem.