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Stochastic Calculus I & II: Course Outline for Stochastic Calculus I (MAT 645)

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes.

Course Contents

The Binomial Asset Pricing Model, Finite Probability Spaces, Lebesgue Measure and the Lebesgue Integral,General Probability Spaces, Independence, Independence of sets, Independence of _-algebras, Independence of random variables, Correlation and independence, Independence and conditional expectation, Law of Large Numbers, Central Limit Theorem.

A Binomial Model for Stock Price Dynamics, Information, Conditional Expectation, An example. Definition of Conditional Expectation, Further discussion of  Partial Averaging, Properties of Conditional  Expectation, Examples from the Binomial Model, Martingales.

Binomial Pricing, Risk-Neutral Probability Measure, Portfolio Process, Self-financing Value of a Portfolio Process, Simple European Derivative Securities, The Binomial Model is Complete. Markov Processes, Different ways to write the Markov property with applications.

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