1 Probability Theory and Stochastic Processes
In this chapter, we introduce the basic notation and fundamental tools of probability theory and measure theory that will be used throughout the subsequent chapters. The section is organized as follows:
- 1.
- probability space and random variables;
- 2.
- modes of convergence;
- 3.
- the Law of Large Numbers (LLN) and the Central Limit Theorem (CLT);
- 4.
- martingales, Markov chains and Brownian motion.
This section mainly follows Introduction to Probability Theory (Tsinghua University), Stochastic Processes (Peking University), Stochastic Analysis (Peking University), Math C218A (UC Berkeley), Math C218B (UC Berkeley) and the book Probability: Theory and Examples by Rick Durrett [3], Sinho Chewi’s notes of MATH C218A, and Introduction to Stochastic Processes [4].
1.1.1 Probability space
1.1.2 Random Variables
1.2 Convergence
1.2.1 Convergence: Almost Sure, in Probability, and in Lp
1.2.2 Weak Convergence
1.2.3 Convergence in Distribution
1.3 The Law of Large Numbers and the Central Limit Theorem
1.3.1 Strong Law of Large Numbers
1.3.2 Central Limit Theorem
1.3.3 Law of the Iterated Logarithm
1.4 Stochastic Processes
1.4.1 Markov Chains
1.4.2 Martingale
1.4.3 Itô Calculus
1.4.4 Stochastic Differential Equations