Contents

1 Lecture 1. Introduction. 1.1 Prices of stocks as functions of time . . . . . . . . . . . . . . . . . . 1.2 Simulated functions of time that look like stock prices . . . . . . . . 1.3 Example of modelling using Random Variables . . . . . . . . . . . . 2 Lecture 2. Random variables. Revision. 2.1 Random Variables and Their Distributions. General. . . . 2.2 Expected value= mean, Variance=( Standard Deviation)2 . 2.3 Common models. Speciﬁc Probability distributions. . . . . 2.4 Realization of Random Variables. Using Excel . . . . . . . 2.5 Expectation of a function of a random variable . . . . . . . 3 Lecture 3. Multivariate distributions. 3.1 Independence . . . . . . . . . . . . . . 3.2 Covariance and Correlation . . . . . . 3.3 Summary of Expectation, Variance and 3.4 Multivariate Normal distribution . . . . . . . . . . . . . . . . . Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 4 4 4 5 8 8 10 10 11 12 12 15 15 16 17 17 18 18 19 20 20 20 22 23 23 23 24 25 26

4 Lecture 4. Tools for distributions. Transforms 4.1 Moment generating functions . . . . . . . . . . 4.2 Characteristic functions . . . . . . . . . . . . . 4.3 Properties of transforms. . . . . . . . . . . . . . 4.4 Moments and moment generating functions. . .

5 Lecture 5. Conditional Expectation 5.1 Conditional Expectation and Conditional Distribution . . . . . . . . 5.2 Properties of Conditional Expectation . . . . . . . . . . . . . . . . 6 Lecture 6. Conditional Expectation as the best predictor 6.1 Expectation as best predictor . . . . . . . . . . . . . . . . . 6.2 Conditional Expectation, Best Possible Predictor with extra mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conditional expectation with many predictors . . . . . . . . 7 Lecture 7. Random Walk and Discrete time Martingales 7.1 Simple Random Walk . . . . . . . . . . . . . . . . . . . . . . 7.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Martingales in Random Walks . . . . . . . . . . . . . . . . . q 7.4 Exponential martingale in Simple Random Walk ( p )Xn . . . 8 Lecture 8. Stopping Times . . . . infor. . . . . . . . . . . . . . . . . . . . . . . .

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9 Lecture 9. Optional Stopping Theorem and Applications 9.1 Optional Stopping Theorem . . . . . . . . . . . . . . . . . . . 9.2 Hitting probabilities in a simple Random Walk- application of tional Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Duration of a game, Eτ -application of Optional Stopping . . .

. . . Op. . . . . .

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10 Lecture 10. Brownian Motion 10.1 Brownian Motion as a Gaussian Process . . . . . . . . . . . . . . . 10.2 Covariance function . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lecture 11. Transition probability density of Brownian motion 11.0.1 Finite dimensional distributions . . . . . . . . . . . . . . . . 11.1 Realizations of Brownian motion . . . . . . . . . . . . . . . . . . . 11.2 Scaling property of Brownian motion. Self-similarity . . . . . . . . . 12 Lecture 12. Processes obtained from Brownian motion 12.1 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Simulation of Brownian motion and related processes . . . . . . . . 13 Lecture 13. Stochastic Calculus. 13.1 Non-diﬀerentiability of Brownian motion . . . . . 13.2 A continuous but no-where diﬀerentiable function 13.3 White noise process . . . . . . . . . . . . . . . . . 13.4 Itˆ Integral. . . . . . . . . . . . . . . . . . . . . . o 13.5 Distribution of Itˆ...