Probability and Stochastic Calculus


The aim of this course is to gain a working understanding of Stochastic Calculus with Brownian motion. To be self contained we cover Probability Theory and a gentle introduction to ideas from Measure Theory and the theory of discrete time Martingales.

Part 1: Probability

1.1 Discrete Probability
1.2 Independence
1.3 Random Variables
1.4 Independent Random Variables
1.5 Mean and Variance
1.6 Bernoulli Trials and the Binomial Distribution
1.7 Distribution Functions and Continuous Random Variables
1.8 The Normal Distribution
1.9 The Log-Normal Distribution

Part 2: Stochastic Calculus

2.1 Some Necessary Theory: Elements of Measure Theory
2.2 Random Variables and Measurability
2.3 Stochastic Processes
2.4 Filtration
2.5 Conditional Expectation and Martingales
2.6 Brownian Motion
2.7 Construction of the Ito Integral
2.8 The Ito Formula
2.9 The Girsanov Theorem
2.10 The Feynman-Kac Formula
2.11 The Fokker-Planck Equation
2.12 Geometric Brownian Motion
2.13 The Ornstein Uhlenbeck Process

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