Stochastic Integration and Differential Equations

It has been 15 years since the first edition of **Stochastic Integration and Differential Equations**, **A New Approach** appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The **new edition** has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and EmeryвЂTMs examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space H^1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it a new approach.

The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery’s examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, a treatment of the Fefferman martingale inequality, and that the dual space of the martingale space Hsub1 can be identified with BMO martingales. Solutions to selected exercises are available at the web site of the author, with current URL http://www.orie.cornell.edu/~protter/books.html.

Booknews

An introduction to the subject, requiring some knowledge of the theory of stochastic processes, including elementary martingale theory. The new approach considers semimartingales as good integrators rather than as the sum of a local martingale and a finite variation process. Examples such as Brownian motion, the Poisson process, and Levy processes are treated in detail. Annotation c. Book News, Inc., Portland, OR (booknews.com)

It has been 13 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though we will no longer call it a new approach . The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises! Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been nearly completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, the more general version of the Girsanov theorem due to Lenglart, the Kazamaki-Novikov criteria for exponential local martingales to be martingales, and a modern treatment of compensators. Chapter 4 treats sigma martingales (important in finance theory) and gives a more comprehensive treatment of martingale representation, including both the Jacod-Yor theory and Emery's examples of martingales that actually have martingale representation (thus going beyond the standard cases of Brownian motion and the compensated Poisson process). New topics added include an introduction to the theory of the expansion of filtrations, and an elementary treatment of the Burkholder-Gundy-Fefferman martingale inequalities. Last, there are of course small changes throughout the book

from The Author Of The Highly Acclaimed A First Course In Real Analysis Comes A Volume Designed Specifically For A Short One- Semester Course In Real Analysis. Many Students Of Mathematics And Those Students Who Intend To Study Any Of The Physical Sciences And Computer Science Need A Text That Presents The Most Important Material In A Brief And Elementary Fashion. The Author Has Included Such Elementary Topics As The Real Number System, The Theory At The Basis Of Elementary Calculus, The Topology Of Metric Spaces And Infinite Series. There Are Proofs Of The Basic Theorems On Limits At A Pace That Is Deliberate And Detailed. There Are Illustrative Examples Throughout With Over 45 Figures.

From the author of the highly acclaimed A First Course in Real Analysis comes a volume designed specifically for a short one-semester course in real analysis. Many students of mathematics and those students who intend to study any of the physical sciences and computer science need a text that presents the most important material in a brief and elementary fashion. The author has included such elementary topics as the real number system, the theory of the basis of elementary calculus, the topology of metric spaces, and infinite series. There are proofs of the basic theorems on limits at a pace that is deliberate and detailed. There are illustrative examples throughout with over 45 figures. From the author of the highly-acclaimed "A First Course in Real Analysis" comes a volume designed specifically for a short one-semester course in real analysis. Many students of mathematics and the physical and computer sciences need a text that presents the most important material in a brief and elementary fashion. The author meets this need with such elementary topics as the real number system, the theory at the basis of elementary calculus, the topology of metric spaces and infinite series. There are proofs of the basic theorems on limits at a pace that is deliberate and detailed, backed by illustrative examples throughout and no less than 45 figures. This volume is designed specifically for a short one-semester course in real analysis. Many students of mathematics and those students who intend to study any of the physical sciences and computer science need a text that presents the most important material in a brief and elementary fashion. The author has included such elementary topics as the real number system, the theory at the basis of elementary calculus, the topology of metric spaces and infinite series. There are proofs of the basic theorems on limits at a pace that is deliberate and detailed. There are illustrative examples throughout with over 45 figures For anyone involved in the physical sciences or computer science, a basic knowledge of real analysis is essential. This book is designed to give a concise treatment of the topic for readers working in those areas. Instead of the lengthy treatments found in most books on the subject, this book presents the most important material in a brief and elementary fashion. 80 illus.