Stochastic Calculus and Financial Applications
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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The Next Steps
Richness of Paths
Localization and It6s Integral
Itos Formula 111
Stochastic Differential Equations
The Diffusion Equation
Arbitrage and Martingales
The Feynmaanac Connection 263
Comments and Credits
Arbitrage and SDEs 153
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a-ﬁeld A-system apply arbitrage arbitrage price argument basic beneﬁt Black—Scholes formula Black—Scholes PDE bounded Brownian bridge calculation coefﬁcients complete the proof conditional expectation conﬁrm consider contingent claim continuous martingale covariance deﬁned deﬁnition denote density derivation diffusion equation distribution dominated convergence theorem Doob’s example EXERCISE fact ﬁltration ﬁnancial ﬁnd ﬁnite ﬁrst note ﬁrst step function gambler geometric Brownian motion Girsanov given gives identity important initial-value problem isometry Ito integral ItO’s formula LÚvy’s linear local martingale martingale property martingale representation theorem mathematical nonnegative portfolio probability measure process deﬁned proposition prove provides Q-martingale random variables random walk reﬂection replicating representation theorem satisﬁes self-ﬁnancing sequence simple random walk solution solve space Speciﬁcally standard Brownian motion stochastic integral stock price strategies submartingale tells transform uniform integrability uniqueness variance wavelet zero