Advanced Stochastic Processes

Lecture Notes

LEC #	TOPICS	LECTURE NOTES
1	Metric spaces and topology	Lecture 1: Metric spaces (PDF)
2	Large deviations for i.i.d. random variables	Lecture 2: Large deviations technique (PDF)
3	Large deviations theory Cramér’s theorem	Lecture 3: Cramér’s theorem (PDF)
4	Applications of the large deviations technique	Lecture 4: Applications of large deviations (PDF)
5	Extension of LD to ℝdand dependent process Gärtner-Ellis theorem	Lecture 5: LD in many dimensions and Markov chains (PDF)
6	Introduction to Brownian motion	Lecture 6: Intro Brownian motion (PDF)
7	The reflection principle The distribution of the maximum Brownian motion with drift	Lecture 7: Brownian motion (PDF)
8	Quadratic variation property of Brownian motion	Lecture 8: Quadratic variation (PDF)
9	Conditional expectations, filtration and martingales	Lecture 9: Filtration and martingales (PDF)
10	Martingales and stopping times I	Lecture 10: Martingales I (PDF)
11	Martingales and stopping times II Martingale convergence theorem	Lecture 11: Martingales II (PDF) Additional materials: Martingale convergence theorem (PDF)
12	Martingale concentration inequalities and applications	Lecture 12: Martigales concentration inequality (PDF)
13	Concentration inequalities and applications	Lecture 13: Talagrand’s concentration inequality (PDF)
14	Introduction to Ito calculus	Lecture 14: Ito calculus (PDF)
15	Ito integral for simple processes	Lecture 15: Ito construction (PDF)
Midterm Exam
16	Definition and properties of Ito integral	Lecture 16: Ito integral (PDF)
17	Ito process Ito formula	Lecture 17: Ito process and formula (PDF)
18	Integration with respect to martingales	Notes unavailable
19	Applications of Ito calculus to financial economics	Lecture 19: Ito applications (PDF)
20	Introduction to the theory of weak convergence	Lecture 20: Weak convergence (PDF)
21	Functional law of large numbers Construction of the Wiener measure	Lecture 21: Tightness of measures (PDF)
22	Skorokhod mapping theorem Reflected Brownian motion	Lecture 22: Reflected Brownian motion (PDF)
Final Exam