Tssp-2024-25 — различия между версиями

Lecture slides and class notes

2024-09-02, lecture 1:

2024-09-09, lecture 2:

2024-09-16, lecture 3: Markov chain: communicating classes. Transient states. Recurrent states.

2024-09-24, lecture 4: Idea of generating function: describe collection of objects as a function and extract information from function. How to extract E(X), E(X^2), E(XY), P(X=3) from a function that generates outcomes. Formal definition of probability generating function and moment generating function.

2024-10-30, lecture 5:

2024-10-07, lecture 6:

2024-10-14, lecture 7: Sigma-algebra is a way to model information, formal definition. Calculating sigma-algebra generated by two events or by discrete random variable. Filtration is a growing sequence of sigma-algebras. Formal definition of conditional expected value with respect to sigma-algebra.

2024-10-21, lecture 8:

2024-12-02, lecture: Girsanov theorem, European option pricing in the Black and Scholes model

Classes

Class video recordings

Maria Kirillova notes

2024-09-06, class 1: First step analysis, 1.1 from StoPro.

More on first step analysis: section 2.7.2 in In2Pro

2024-09-13, class 2: First step analysis, 1.4 from StoPro.

2024-09-20, class 3: Classification of states in Markov chain, communicating classes, 3.1ab from StoPro.

2024-09-27, class 4: Generating functions: standard normal distribution, chi-squared with 1 degree of freedom.

2024-10-04, class 5: Calculating probability limit using LLN. Intuition behind probability limit: unique forecast that is "arbitrary good" for almost all X_n. Probability limit of max and min. Probability limit is a random variable. Probability limit of iid sequence does not exist.

2024-10-11, class 6: Two more limits (in probability and in L2), conditional expected value in uniform case, conditional expected value with joint density.

2024-10-18, class 7: Calculation of sigma-algebra generated by random variable. Calculation of expected value wrt to sigma-algebra.

2024-11-01, class 8: Checking that a process is a martingale, 9.1abcd, 9.9 from StoPro.

2024-11-08, class 9: Poisson point process

2024-11-15, class 10: Doob's optional stopping time theorem in discrete time

2024-11-22, class 11: Calculating E, Cov, Var for Wiener process, (W_t) and (W_t^2 - t) are martingales

2024-11-29, class 12: Ito's integral for a piece-wise constant process, Ito's lemma, martingale condition

2024-12-06, class 13: more Ito's lemma and stochastic integral properties.

2024-12-13, class 14: binomial tree, finding risk neutral probabilities, European and American call option.