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