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Stochastic Differential Equations

Author: chu-ching Haung
Url: http://diffusion.cgu.edu.tw:8080/forgetmenot/book/StocCal_part2.pdf
Format: Pdf
Year: 2007
Category: Stochastic differential equation
Pages: 130
Clicks: 58

This lecture is designed for the persons who want to develop p rofessional skill in stochastic cal- culus and its application to problems in finance but don’t inv olve in too much mathematics. As well-known, the more experience we have with mathematical s kills (especially real analysis and probability), the more piece we can get. Nevertheless, the c ourse will move quickly and students can expect to acquire tools that are deep enough and rich enou gh to be relied upon throughout their professional careers.The course begins with basic concepts and theory of probabil ity space, needed in this lecture. This material is used to motivate the theory of continuous time stochastic processes, especially Brownian motion. The con struction of Brownian motion is given in detail, and enough material on the subtle properties of Br ownian paths is developed so that the student should evolve a good sense of when intuition can b e trusted and when it cannot. The course then takes up the Itô integral and aims to provide a development that is honest and complete without being pedantic. With the Itô integral in ha nd, the course focuses more on models. Stochastic processes of importance in Finance and E conomics are developed in concert with the tool. The financial notion of replication is develop ed, and the Black-Scholes PDE will be introduced with its importance. At the last part of this co urse, we introduce 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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