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)

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)

Extension of LD to d and 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)

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)

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)

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)

Functional law of large numbers

Construction of the Wiener measure

Lecture 21: Tightness of measures (PDF)

Skorokhod mapping theorem

Reflected Brownian motion

Lecture 22: Reflected Brownian motion (PDF)
Final Exam