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