Bálint Tóth:

Stochastic Differential Equations (BMETE95MM08)

Spring 2017

REQUIREMENTS (TÁRGYKÖVETEKMÉNYEK) – in Hungarian

TIMETABLE:

LECTURES:

WED: 12:15-14:00,   H 46
THU:  10:15-11:00,   H 67 

PROBLEM CLASS / TUTORIAL:

THU:  11:15-12:00,   H 67 

OFFICE HOUR:

Thu:    14:00-15:00,   H 510

        PRELIMINARY SCHEDULE:

Preliminary schedule of the course

LECTURE NOTES (hand written, downloadable, these notes cover essentially all material done in class):

1. Brownian motion 1: motivation, phenomenological description, construction

3. Brownian motion 2: construction

2. Brownian motion 3: distributional and path-wise properties

4. Brownian motion 4: quadratic variation

5. Brownian motion 5: reflection principle

6. Filtrations, stopping times, martingales, Markov processes (recap)...

7. Ito calculus 1: the Ito integral

8. Ito calculus 2: Ito’s formula

9. Stochastic differential equations: strong solution, existence and uniqueness

10. Diffusions 1: infinitesimal generator, Dynkin’s formula

11. Diffusions 2: Kolmogorov’s bw and fw equations

12. The Bessel-Squared and the Bessel process

13. Diffusions and related elliptic PDEs (Laplace, Poisson, Helmholtz with Dirichlet boundary conditions)

14. Diffusions and related parabolic PDE (Heat eq, Kolmogorov backward and forward eqs, Feynman-Kac formula)

15. Diffusions 3: Feller property, contraction semigroups, Hille-Yosida thm, Feynamn-Kac formula

16. Change of measure and Girsanov’s theorem

        BOOKS, ADDITIONAL READING:

        There are many excellent books on the subject of this course. Here are some examples:

        B. Oksendal: Stochastic Differential Equations: An Introduction with Applications. Sixth Edition. Springer, 2010

        I, Karatzas, S.E. Shreve: Brownian Motion and Stochastic Calculus. Second Edition. Springer, 1991

        P. Moerters, Y. Peres: Brownian Motion. Cambridge University Press, 2010

T.M. Liggett: Continuous Time Markov Processes. AMS, 2010

PROBLEM SETS, HOME WORK ASSIGNMENTS:

Problem sheets (downloadable)	Homework assignements	Due date	Solutions (downloadable after hw due dates)
1. Brownian motion	1.1, 1.2, 1.5, 1.13 1.6, 1.8, 1.10, 1.12	16 Feb 02 Mar	solutions
2. Martingales	2.4, 2.5, 2.6, 2.7	09 Mar	solutions
3. Ito calculus	3.1, 3.2, 3.3, 3.4 3.5, 3.7, 3.10, 3.12	16 Mar 30 Mar	solutions
4. Stochastic differential equations	4.2, 4.3, 4.4, 4.5	06 Apr	solutions
5. Diffusions	5.1, 5.3, 5.4, 5.6	20 Apr	solutions
6. Semigroups, Hille-Yosida, Feynman-Kac	to be announced	04 May	solutions
7. Girsanov’s formula	7.2, 7.3, 7.4, 7.8	11 May	solutions

EXAM QUESTIONS FROM EARLIER YEARS:

                   2013: questions solutions