# Stochastic Calculus I & II: Course Outline for Stochastic Calculus-II (MAT 646)

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

First Passage Time, is almost surely finite,The moment generating function for, Expectation of the Strong Markov Property, General First Passage Times , Example: Perpetual American Put, Difference Equation, Distribution of  First Passage Times, The Reflection Principle. Radon-Nikodym Theorem, Radon-Nikodym Martingales, The State Price Density Process, Stochastic Volatility Binomial Model ,Another applicaton of the Radon-Nikodym Theorem. Law of a Random variable, Density of a Random Variable ,Expectation, Two random variables, Marginal Density, Conditional Expectation, Conditional Density Multivariate Normal Distribution, Bivariatenormal  distribution, MGF of jointly normal random variable. Discrete-time Brownian Motion,The Stock Price Process, Remainder of the Market, Risk-Neutra l Measure, Risk-Neutral Pricing, Arbitrage, Stalking the Risk-Neutral Measure, Pricinga European Call. Symmetric Random Walk, The Law of Large Numbers, Central Limit Theorem, Brownian Motion as a Limit of Random Walks, Brownian Motion, Covariance of Brownian Motion, Finite-Dimensional Distributions of Brownian Motion, Filtration generated by a Brownian Motion, Martingale Property, The Limit of a Binomial Model Starting at Points Other Than 0, Markov Property for Brownian Motion, Transition  Density, First PassageTime.

Brownian Motion, First Variation, Quadratic Variation, Quadratic Variation as Absolute Volatility. Construction of  the It ˆo Integral, It ˆo integral of an elementary integrand, Properties of the It ˆo integral of an elementary process, It ˆo integral of a general integrand ,Properties of  the (general) It ˆo integral Quadratic variation of an It ˆo integral It ˆo’s formula for one Brownian motion, Derivation of It ˆo’s formula, Geometric Brownian motion, Quadratic variation of geometric Brownian motion,Volatility of Geometric Brownian motion  First derivation of the Black-Scholes formula, Mean and variance of the Cox-Ingersoll-Ross process, Multidimensional Brownian Motion, Cross-variations of Brownian motions, Multi-dimensional It ˆo formula. Stochastic Differential Equations, Markov Property, Transition density, The Kolmogorov Backward Equation, Connection between stochastic calculus and KBE, Black-Scholes, Black-Scholes with price-dependent volatility. Conditional expectations under, Risk-neutral measure, Martingale Representation Theorem, A hedging application, d-dimensional Girsanov Theorem, d-dimensional Martingale Representation Theorem, Multi-dimensional market model.