Brownian Motion and Stochastic Calculus

I. KARATZAS & S.E. SHREVE (1988) Brownian Motion and Stochastic Calculus. Volume 113 in the series "Graduate Texts in Mathematics", Springer-Verlag, New York, Heidelberg & Berlin. 470 pages. Subsequent Printings: 1991, 1994, 1996, 1997, 1999, 2000, 2003, 2005, 2006. Chinese Edition: World Publishing Corporation, 1990, 1995, 2006. Japanese Translation: Springer-Verlag Tokyo, 2002.
From the backflap of this book:

"This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes, who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization).

This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.



Methods of Mathematical Finance

I. KARATZAS, I. & S.E. SHREVE (1998) Methods of Mathematical Finance. Volume 39 in the series "Applications of Mathematics", Springer-Verlag, New York, Heidelberg, and Berlin. 407 pages. Subsequent Printings: 1999, 2001. Chinese Edition: World Publishing Corporation, 2006.

From the backflap of this book:

"This monograph is a sequel to the book 'Brownian Motion and Stochastic Calculus' by the same authors. Within the context of Brownian-motion-driven asset prices, it develops contingent claim pricing and optimal consumption/investment in both complete and incomplete markets. The latter topic is extended to a study of equilibrium, providing conditions for existence and uniqueness of market prices which support trading by several heterogeneous agents. Although much of the incomplete-market material is available in research papers, these topics are treated for the first time in a unified manner. The book contains an extensive set of references and notes describing the field, including topics not treated in the text.

This monograph will be of interest to researchers wishing to see advanced mathematics applied to finance. The material on optimal consumption and investment, leading to equilbrium, is addressed to the theoretical finance community. The chapters on contingent claim valuation presents techniques of practical importance, especially for pricing exotic options.


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