Advanced Probability and Stochastic Processes

主要工作两个部分

（1）完成作业（一般来说，每周一次，看你的功力，一般不超过5个题）

（2）作业简单讲解和答疑（每周不超过3小时，通过电话和网络， 100元每小时）

真学霸，一周工作可能需要5个小时，收入不少于500. 可以开月薪，可以开到~3000，可商量，也可开周薪, 每周不少于 500, 具体可商量。

联系 fin.math.student@gmail.com，我给你发作业题样题，能答题，即可马上开始工作。

feel free to forward this message to anyone you feel capable of doing this part-time job.

以下为课程大纲 -

Syllabus

I Lecture 1. Measure theory and probability space

Measurable space, -algebra

Axioms of probability and probability space

Random variables and random vectors

Representations of random variables and random vectors

Commonly used distributions

Lebesgue integration theory and expectation

Fubini theorem

I Lecture 2. More on random variables and random vectors

Integral convergence theorems

Important inequalities

Lp space

Convergence of random variables

Aggregation and projection of random variables

Simulation of single variate random variable

I Lecture 3. Series of independent random variables

Independent events, independent -algebras, and independent random

variables - theorem

Sum of independent random variables

Kolmogorov inequality

Kolmogorov 3-series theorem

I Lecture 4. Law of large numbers

Weak law of large numbers

Borel-Cantelli lemmas

Kolmogorov's 0-1 law

Strong law of large numbers

Glivenko-Cantelli lemma

I Lecture 5. Characteristic functions and weak convergence

Characteristic functions

1Topics are subject to change.3

Properties of characteristic function

Fourier inversion formula

Convergence in distribution and weak convergence

Levy-Cramer continuity theorem

Bochner's theorem

Central limit theorem

Lindeberg-Feller theorem

Option pricing via characteristic functions

I Lecture 6. More on weak convergence

Levy's theorem

Stable laws

Innitely divisible distributions

Compound Poisson distributions

Levy-Khintchine formula

I Lecture 7. Dependent structure and copulas

Sklar theorem

Frechet upper and lower copulas

Archimedean copulas

Simulation of copulas

How to choose a copula

Pricing of multiasset options, e.g., basket option, spread option

I Lecture 8. Martingale theory I

Conditional probability and conditional expectation

Properties of conditional expectation

Uniform integrability

Stochastic processes in general

Martingale, submartingale, supermartingale

Doob's inequalities

I Lecture 9. Martingale theory II

Martingale convergence theorems

Martingale central limit theorem

Predictable processes4

Doob decomposition theorem

Stopping times

Martingale transformation and discrete stochastic integral

Optional stopping theorem

I Lecture 10. Markov processes

Makov property and strong Markov property

Transition matrix, transition density

Chapman-Kolmogorov equation

The generator and the innitesimal generator

Markov martingales

Dynkin's formula

I Lecture 11. Time series analysis I

The lag operator

ARMA processes

Autocovariance and autocorrelation functions

Covariance stationary processes

Autocovariance-generating function

Spectral density function

Spectral representation theorem

Stationarity and ergodicity

The Wold theorem

I Lecture 12. Time series analysis II

Unit root processes

Cointegration

Volatility clustering and GARCH process