Instructor: Balazs Rath

Rick Durrett: Probability: Theory and Examples (PTE): CLICK

My hand-written, scanned lecture notes:

Sept 7 (measure theoretic probability, sigma-algebras, conditional expectation for discrete and continuous rv's): PDF (pages 0-6)

 

Sept 8 (abstract conditional expectation, properties, filtration, martingale, discrete stoch. integral): PDF (pages 6-13)

 

Sept 15 (simple random walk and related martingales, casino, optional stopping thm, discrete Doob-Meyer decomposition): PDF (pages 14-21)

 

Sept 21 (square of SRW, SRW hitting probabilities, multivariate normal distribution): PDF (pages 22-29)

 

Sept 22 (continuous-time stoch. proc., Markov proc., Gaussian proc., Brownian motion ): PDF (pages 30-36)

 

Sept 28 (Paul Levy's construction of Brownian Motion): PDF (pages 37-44)

 

Sept 29 (properties of B.M., martingales derived from B.M., Stieltjes integral, quadratic variation, mutual variation): PDF (pages 45-52)

 

Oct 5 (properties of quadratic/mutual var., strong Markov property, reflection principle): PDF (pages 52-59)

 

Oct 6 (simple predictable process, Ito isometry, def of Ito integral, Ornstein-Uhlenbeck process): PDF (pages 60-66)

 

Oct 12 (Ito and Stieltjes integral, properties of Ito integral, examples, the case of deterministic integrand): PDF (pages 67-71)

 

Oct 13 (deterministic integrand produces Gaussian process, Ito int is continuous martingale if integrand is simple predictable): PDF (pages 72-75)

 

Oct 19 (Ito int is continuous martingale, submartingale inequality): PDF (pages 73-79)

 

Oct 20 (O-U process again, calculation of Ito integrals, quadratic variation of Ito integrals, def of Ito process): PDF (pages 80-86)

 

Oct 26 (chain rule of regular calculus, statement and proof of Ito's formula): PDF (pages 87-93)

 

Oct 27 (the difference between deterministic and random integrand, rules of Ito calculus, Brownian bridge): PDF (pages 94-100)

 

Nov 2 (quadratic/mutual variation of Ito processes, Ito integral w.r.t. Ito proc., Ito formula for Ito proc. ): PDF (pages 101-108)

 

Nov 3 (heu. proof of Ito formula for Ito proc., stoch. integration by parts, examples and counterexamples): PDF (pages 109-115)

 

Nov 9 (Ito calculus examples, time-dependent Ito formula, a remark about optional stopping theorem): PDF (pages 116-122)

 

Nov 10 (an example of the matringale representation thm, Ito process driven by d-dim B.M., multi-variable Ito formula): PDF (pages 123-130)

 

Nov 16 (heu. proof of multi-variable Ito formula, harmonic functions and martingales, Paul Levy's characterization of B.M.): PDF (pages 131-136)

 

Nov 23 (exit time from a ball, a starnge process which has zero drift but not a martingale, 3d B.M. never hits origin): PDF (pages 137-144)

 

Nov 24 (O-U process solves the Langevin equation, stochastic exponential, geometric Brownian motion): PDF (pages 145-148)

 

Nov 30 (SDE for Brownian bridge, "repulsive" Langevin equation and hitting times, Bessel process def, stochastic logarithm): PDF (pages 149-155)

 

Dec 1 (Ito diffusion processes, transition density function of geometric B.M., solution of general linear SDE): PDF (pages 156-163)

 

Dec 7 (Bessel proc. hitting probabilities, phase transition of Bessel proc., CIR process, expected hitting time for Bessel proc.): PDF (pages 164-171)

 

Dec 8 (stationary distribution of Ito diffusion processes, example: CIR, Stochastic logistic equation): PDF (pages 172-179)