1. R. Durrett, Probability: Theory and Examples 4th Edition, Cambridge University Press, 2013. Math 668 will cover approximately Chapters 3.1-3.3, 5.1-5.3 (martingales), 6.1-6.8 (Markov chains), with occasional forays into Oksendal for variety. Oksendal will be used more in 669.
2. B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2010. (Earlier editions should work too.)

1. Friday Jan 9: 3.3.9 (note: x^2 is not bounded!), 3.3.10
2. Friday Jan 16: 5.1.2, 5.1.3, 5.1.9
3. Friday Jan 23: 5.2.6, 5.2.9 and apply it to Y_n = e^(t Z_n)/mgf(t), Z_n Bernoulli p (this is closely related to Wald's identity for stopping a sum at a random time, which will follow from the optional stopping theorem) (some solutions)
4. Friday Jan 30: 5.2.4, 5.3.13
5. Friday Feb 6 Midterm
6. Friday Feb 13: plot the 8 trajectories of the chromatic number martingale with respect to the vertex exposure filtration, using p=1/2 for including edges and 3 vertices, and verify that increments are bounded by 1, sketch of solution
7. Friday Feb 20: 6.2.4, 6.2.9 note: 6.2.9 is the DAG model for exchangeability of X_i. Use the Beta density to get the integrals to compute the transition function. The transition probabilities are given by P(S_{n+1}-S_{n}=1 | X_1, ..., X_n) = ( [S_{n}+n]/2 + 1 )/ (n+2), or in our books notation p_n(i,i+1)=([i+n]/2 + 1)/(n+2), p_n(i,i-1) = ([n-i]/2 + 1)/(n+2) careful with notation.
8. Friday Feb 27: 6.3.10 (apply to A={a,b} for symmetric random walk to get expected time-to-ruin), 6.3.11
9. Friday Mar 6: 6.5.10 and apply to symmetric rw on Z, 6.6.1.

                       
    January 2015    
Su Mo Tu We Th Fr Sa
             1  2  3 
 4  5  6  7  8  9 10	3.3
11 12 13 14 15 16 17 	5.1
18 19 20 21 22 23 24 	5.2, Monday holiday 
25 26 27 28 29 30 31 	5.2
                     
    February 2015   
Su Mo Tu We Th Fr Sa
 1  2  3  4  5  6  7 	5.3
 8  9 10 11 12 13 14 	5.7 Optional Stopping Theorem 
15 16 17 18 19 20 21 	6.1-6.2, Bad Gibbs Sampler 
22 23 24 25 26 27 28	6.3-6.4
                     
                     
     March 2015     
Su Mo Tu We Th Fr Sa
 1  2  3  4  5  6  7 	6.5
 8  9 10 11 12 13 14 	6.6, 6.8
15 16 17 18 19 20 21 	final exams 
22 23 24 25 26 27 28 
29 30 31