The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It(tm) integral and aims to provide a development that is honest and complete without being pedantic. With the It(tm) integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers.
The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It¿ integral and aims to provide a development that is honest and complete without being pedantic. With the It¿ integral in hand, the course focuses more on models.
Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
This book is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance. The Wharton School course that forms the basis for this book is designed for energetic students who have had some experience with probability and statistics but have not had ad vanced courses in stochastic processes. Although the course assumes only a modest background, it moves quickly, and in the end, students can expect to have tools that are deep enough and rich enough to be relied on throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more de manding development of continuous-time stochastic processes, especially Brownian motion. The construction of Brownian motion is given in detail, and enough mate rial on the subtle nature of Brownian paths is developed for the student to evolve a good sense of when intuition can be trusted and when it cannot. The course then takes up the Ito integral in earnest. The development of stochastic integration aims to be careful and complete without being pedantic. A graduate level methematical introduction to stochastic calculus using financial applications as examples. Starts with the discrete stochastic process then quickly moves on to continuous stochastic process. Suggested prerequisite courses are calculus I, II, and III (multivariate calculus), ordinary differential equations (ODE), partial differential equations (PDE), and probability and measure theory. A prior course in stochastic process is not necessary. Some readers on Amazon.com have suggested that real analysis (advanced calculus) may also be a prerequisite. Author is a professor of statistics at University of Pennsylvania and this book is used in his class for advanced MBA (or Finance PhD) students at Wharton. "The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous-time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should evolve a good sense of when intuition can be trusted and when it cannot. The course then takes up the Ito integral and aims to provide a development that is honest and complete without being pedantic."--Jacket "This book is designed for students who want to develop professional skills in stochastic calculus and its application to problems in finance. The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly, and in the end, students can expect to have the tools that are deep enough and rich enough to be relied on throughout their professional careers." Stochastic calculus has important applications to mathematical finance. This book will appeal to practitioners and students who want an elementary introduction to these areas. From the reviews: "As the preface says, This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract. This is also reflected in the style of writing which is unusually lively for a mathematics book." --ZENTRALBLATT MATH "This book offers a course which begins with a simple analysis of gambling games, which material is used to motivate the theory of martingales, and then takes up the development of continuous time stochastic process, especially Brownian motion, so to develop in the reader a sense of when intuitition can be trusted and when it cannot. The course then takes up the Ito integral." Designed as an introduction to stochastic calculus and its application to mathematical finance, Steele gives the reader tools that are deep and rich enough to be used throughout their professional careers, with a focus on mathematical finance and economics. Random Walk and First Step Analysis First Step Analysis Time and Infinity Tossing an Unfair Coin Numerical Calculation and Intuition First Steps with Generating Functions Exercises First Martingale Steps Browian Motion Martingales: The Next Steps Preface This book is designed for students who want to develop professional skil in stochastic calculus and its applicatio to problems in finance.
