Stochastic Calculus and Financial Applications

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 302 pages
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.
 

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Contents

First Martingale Steps
11
Brownian Motion
29
The Next Steps
43
Richness of Paths
61
ItÚ Integration
79
Localization and It6s Integral
95
Itos Formula 111
110
Stochastic Differential Equations
137
The Diffusion Equation
169
Representation Theorems
191
Girsanov Theory
213
Arbitrage and Martingales
233
The Feynmaanac Connection 263
262
Mathematical Tools
277
Comments and Credits
285
Bibliography 293
292

Arbitrage and SDEs 153
152

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