Stochastic Differential Equations -- spring semester 2019


QUESTION SESSION before the 2nd exam: Tuesday, the 11th of May 2019, from 10:00, in room H601.

This was the midterm and here are the solutions.
This was the makeup midterm and here are the solutions.
This was the 2nd makeup midterm and here are the solutions.
This was the 1st exam and here are the solutions.
This was the 2nd exam and here are the solutions.


Here is an exam from last year. Questions this year will be similar in flavour.

Results of the semester here.
Explanation of the results:

• the "raw score" of homeworks is the number of "correctly solved" exercises plus 0.8 times the number of "solved" exercises. The "homework score" is this raw score times 2.5 (but no more than 20). This way, solving 8 homeworks correctly is enough to get the maximum score of 20. (The code of studies only allows taking 70% of the homeworks into account, since there was no option to replace them.)
• The "midterm score" is 1.5 times the number of points reached on the midterm or the makeup midterm (but no more than 30). This way, reaching 20 points was enough to get the maximum score of 30.
• The signature is given if at least 8 homeworks were submitted and the midterm score is at least 12 (meaning that at least 8 points were reached).

Course code: BMETE95MM08
Course code for the lecture: T0
Course code for the practice: T1
Official couse data page (in hungarian): TE95MM08
Classes:

• Wednesday 12:15-14:00, room H406
• Thursday 10:15-12:00, room H607

Lecturer: Tóth Imre Péter. Lectures and practice classes are not strictly separated in schedule.


LECTURE NOTES (hand written by Bálint Tóth, 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 NOT DISCUSSED AND NOT NEEDED
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 NOT DISCUSSED AND NOT NEEDED
16. Change of measure and Girsanov’s theorem


PROBLEM SETS, HOMEWORK ASSIGNMENTS

Problem sheets	Homework assignments	Due date	Solutions
1. Brownian motion	1.10, 1.12, 1.14, 1.18	28 Feb	here
2. Martingales	2.9, 2.10, 2.11	04 April	here
3. Ito intergral, Ito formula	3.15, 3.16	08 May	here
4. Stochastic differential equations, Dynkin formula, Girsanov theorem	4.3, 4.7, 4.13	on the exam	here - except for the hand-in homeworks

REQUIREMENTS There will be regular homework assignments and a written midterm exam during the study period, and a written final exam in the exam period.

• Signature: To get the signature,
  • at least 70% of the assigned homeworks have to be submitted, with a real attempt to solve them,
  • on the midterm exam at least 40% of the total score must be reached. (Since the midterm is graded on a scale of 20, this means 8 points.)
• Final grade: Homework assignments are worth 20% of the total score. The midterm will be worth 30%, while the final exam is worth 50%.

Homework grading rules: The solution to each homework is evaluated with a "code" with the following meaning:

• 3 means the solution is correct.
• 2 means the solution is essentially correct, but there is some error.
• 1 means the student started on a good track, but could not solve the problem.
• 0 means the solution is completely wrong.

In the end, these codes are translated into scores using the following rule:

• "correct" (code 3) is worth 1 score,
• "essentially correct" (code 2) is worth 0.8 scores,
• "started" and "wrong" (codes 1 and 0) are worth no scores at all.

The final grade, as a function of the total score (on the scale 0-100) is given by the following table:

0-39 scores	1
40-54 scores	2
55-69 scores	3
70-84 scores	4
85-100 scores	5

MIDTERM EXAM

• Midterm exam: Wednesday, the 10th of April 2019, from 16:00 (on week 9) in room H607
• Retake midterm exam: Monday, the 13th of May 2019, from 14:00 (on week 14) TIME CHANGE!!!in room T605
• Second retake midterm exam: Tuesday, the 21st of May 2019, from 09:00, room K174 (in the "Delayed completion period") Students are only allowed to participate in the second retake midterm exam if they have participated in the midterm exam or its first retake.