Stochastic Calculus and Financial Applications / Edition 1

Stochastic Calculus and Financial Applications / Edition 1

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by J. Michael Steele
     
 

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 shastic 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

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Overview

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 shastic 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 shastic 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.
Shastic processes of importance in Finance and Economics are developed in concert with the tools of shastic 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.

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Product Details

ISBN-13:
9781441928627
Publisher:
Springer New York
Publication date:
12/01/2010
Series:
Stochastic Modelling and Applied Probability Series, #45
Edition description:
Softcover reprint of hardcover 1st ed. 2001
Pages:
302
Product dimensions:
0.65(w) x 9.21(h) x 6.14(d)

Table of Contents

Random Walk and First Step Analysis
• First Martingale Steps
• Brownian Motion
• Martingale—Next Steps
• Richness of Paths
• Itô Integration
• Localization and Itô's Integral
• Itô's Formula
• Shastic Differential Equations
• Arbitrage and SDE's
• The Diffusion Equation
• Representation Theorems
• Girsanov Theory
• Arbitrage and Martingales
• The Feynman-Kac Connection

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