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The oral exam is distributed over 3 days (starting at 9:00 am), that is

4. June, Monday: Ullev?l, room Hurricane,

5. June, Tuesday: Blindern, NHA, room UE32

6. June, Wednesday: Blindern, NHA, room UE32

The exam procedure is as follows: The exam takes 45 min. and essentially consists of two parts:

1. Talk/ presentation of a topic of free choice about stochastic analysis. The title and topic of the presentation are supposed to be communicated to me (by e-mail) latest 2 days before the exam and approved by myself. The length of the talk is limited to about 20 minutes and the form of the presentation is up to the candidate (blackboard, beamer,...).

2. General questions about stochastic analysis. 

The lecture notes can be found on the website of our course.

A set of exam relevant topics and exercises can be downloaded here: topics, exercises

The...

Our last lesson is supposed to be on Wednesday, 16. May, 10:15-12:00 !

In the last weeks (18., 19., 25., 26. April, 2., 3., 9. May) we discussed some other central results of stochastic analysis as e.g. Dynkin?s formula, Girsanov?s theorem and the Feynman-Kac representation formula, which gives a probabilistic representation of solutions to Cauchy problems and which can be used for the construction of solutions to certain partial differential equations (see chapters 7 and 8 in ?ksendal). Further, we also studied the Hamilton-Jacobi-Bellman equation in connection with stochastic control problems. In our last lesson on Wednesday, 16. May we aim at finishing chapter 11 on stochastic control theory in the book of ?ksendal.

In our last lessons (4., 5., 11., and 12. April) we completed the proof of our main result on stochastic linear filtering theory (Ch. 6 in the book of ?ksendal). Further, we studied the (strong) Markov property of Ito-diffusions and introduced the concept of a generator of such processes (Ch.7 in ?ksendal).

Next week (18., 19. April) we aim at using the concept of a generator to derive the famous Dynkin formula, which can be e.g. applied to the study of exit times. In addition, we also want to discuss a central result in stochastic analysis, that is Girsanov?s theorem, which has a variety of important applications to e.g. non-linear filtering theory or mathematical finance (Ch. 7 in ?ksendal).

In our last lessons (28. Feb., 1., 14. and 15. March) we discussed the construction of strong solutions to SDE's (Ch. 5 in the book of ?ksendal). Further, as an application of SDE theory we studied stochastic filtering theory (Ch. 6 in ?ksendal).

Our plan for next week (21., 22. March) is to complete the proof of the main result in connection with linear filtering (Kalman-Bucy filter).

In our course the exam is supposed to be oral !

The date for the oral exam is not fixed yet.  

Keith Zhou is the elected student representative (keithzhousiyuatgmail.com) of our course !

Last time (14. and 15. Feb.) we were concerned with the construction of the Ito integral with respect to the Brownian motion (see Sect. 3 in ?ksendal). In addition, we also discussed Ito?s Lemma (or Ito?s formula), which can be considered a chain rule for the Brownian motion and which is an important result in stochastic analysis (see Sect. 4 in ?ksendal). Next week (21. and 22. Feb.) we aim at proving the martingale representation theorem for (square integrable Brownian) martingales (sect. 4), which has many interesting applications (e.g. to stochastic control theory or mathematical finance). 

In our last lessons (31. jan., 1. feb.) we finished our crash course on basic notions and results from the theory of stochastic processes (see ch. 2 in the book of B. ?ksendal). Further, we also proved the existence of Brownian motion by using N. Wiener?s arguments from the 1920ties.

On 7. and 8. feb. we aim at discussing the construction of Ito-integrals (see ch. 3 in ?ksendal).  

lecture notes: part1, part2, part3, part4, part5, part6, part7, part8, part9, part10, part11

exercises: Exercises1, Exercises2, Exercises3, Exercises4, Exercises5, Exercises6, Exercises7, Exercises8, Exercises9, Exercises10, Exercises11

solutions: Ex1Prob5, Ex2Prob5, Ex3Prob25, Ex3Prob4,  Ex4Prob15, Ex4Prob6, Ex5Prob2, Ex5Prob56, Ex6Prob1, Ex6Prob345, Ex7Prob25, Ex8Prob1356, Ex9Prob6

compendium measure theory: compendium

NB ! Det blir ingen (!) undervisning neste uke, dvs. p? onsdag, 24. og p? torsdag, 25. januar !

Vi fortsetter med repetisjon av grunnleggende konsept og resultater fra sannsynlighetsteori p? onsdag, 31. jan.

NB ! There will be no (!) lesson next week, that is on Wednesday, 24. and Thursday, 25. January.

We will continue with our review of basic concepts and results from probability theory and the theory of stochastic processes on Wednesday, 31. January.

Vi starter opp med kurset torsdag, 18. januar, kl. 10:15-12:00, NHA, rom UE26 !

We start with the course on Thursday, 18. January, 10:15-12:00, NHA, room UE26 (so there is no lesson on 17th of January) !

Bernt ?ksendal: Stochastic Differential Equations, 2013. Springer. (6th Edition Corrected Printing).

Chapters:  

1.

2.

3. Sections 3.1, 3.2 and 3.3 until page 35 "A comparison of Ito and Stratonovic integrals"

4.

5. From Example 5.3.2 is self-read (pages 75-76)

6.

7. Section 7.1, 7.2 (except page 122-123, "Hitting distributions..."), 7.3, 7.4. Section 7.5 is self-read

8. Section 8.1, 8.2, 8.6

If time permits, we will also study the chapters

11.

12.

Appendix B

In addition, all given exercises.