随机过程笔记
Notes on Stochastic Processes
Kiyoshi Igusa
December 17, 2006
英文版


Contents
Syllabus
0. Linear equations
(a) linear differential equations in one variable
(b) Kermack-McKendrick epidemic model
(c) first order equations
(d) linear difference equations
1. Finite Markov chains
(a) definitions
(b) communication classes
(c) classification of states: transient and recurrent
(d) periodic chains
(e) invariant distributions
(f) return time
(g) substochastic matrix
(h) Leontief model
2. Countable Markov chains
(a) extinction probability
(b) random walk
(c) Stirling’s formula
3. Continuous Markov chains
(a) making time continuous
(b) Poisson to Markov
(c) definition of continous Markov chain
(d) probability distribution vector
(e) equilibrium distribution and positive recurrence
(f) birth-death chain
4. Optimal stopping time
(a) basic problem
(b) basic solutions
(c) cost function
(d) discount payoff
5. Martingales
(a) intuitive description
(b) conditional expectation
(c) -filtration
(d) optimal sampling theorem
(e) martingale convergence theorem
6. Renewal
(a) renewal theorem
(b) age of current process
(c) M/G/1-queueing
7. (Time) Reversal
(a) basic equation
(b) reversible process
(c) symmetric process
(d) statistical mechanics
8. Brownian motion
(a) definition
(b) martingales and L´evy’s theorem
(c) reflection principle
(d) fractal dimension of Z
(e) heat equation
(f) recurrence and transience
(g) fractal dimension of path
(h) drift
9. Stochastic integration
(a) idea
(b) discrete integration
(c) integration with respect to Brownian motion
(d) Itˆo’s formula
(e) continuous martingales
(f) Feynman-Kac
(g) Black-Scholes
References