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NB ! The exam in MAT4720 is in written form and will be on Friday, 7. December, 14:30 (4 hours).
The lecture notes for MAT4720 can be found on the website of our course.
A set of exam relevant topics and exercises can be downloaded here: topics, exercises
The pensum of the course comprises the chapters 2 to 9 in my lecture notes (up to chapter 9.2, in ch. 9.2 only Th. 9.7) or see the corresponding topics in the book of Øksendal (or Cohen, Elliott).
Allowed aids for the examination: none.
Our last lesson is supposed to be on Thursday, 22. Nov., 10:15-12:00 !
On Thursday we aim at discussing Kolmogorov´s backward equation (see Ch. 8 in Øksendal) and Exercises 8 (and Exercises 9, if time permits).
solutions to the assignment: solutions
In our last lessons (17., 18., 24., 25, 31. Oct. and 1., 7., 8. Nov.) we completed the proof of our main result on stochastic linear filtering theory (Ch. 6 in the book of Øksendal). Further, we studied the (strong) Markov property of Ito-diffusions (Ch. 7 in Øksendal).
Next time (14., 15. Nov.) 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 Øksendal).
MandatoryAssignment
Deadline for the electronic (!) submission:
Thursday, 1. November, 14:30 !
In our last lessons (3., 4., 10. and 11. Oct.) we discussed the martingale representation theorem and the construction of strong solutions to SDE's (Ch. 4 and 5 in the book of Øksendal). As an application of SDE theory we aim at studying next week (17. and 18. Oct.) stochastic filtering theory (Ch. 6 in Øksendal).
The student representatives of our course are
1. Åsmund H. Sande (aasmunhs[at]student.matnat.uio.no)
2. Alexandra Filion (alexandra_filion[at]hotmail.com)
Last time (12., 13., 26. and 27. Sept.) we were concerned with the construction of the Ito integral with respect to the Brownian motion (see Sect. 3 in Øksendal). In addition, we also discussed properties of such stochastic integrals (e.g. existence of a continuous version, martingale property,...) and Ito´s Lemma (or Ito´s formula), which can be considered a chain rule for the Brownian motion and which is a basic result in stochastic analysis (see Sect. 4 in Øksendal). Next week (3. and 4. Oct.) 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).
Problems to Exercises 3/Exercises 4 will be presented on Thursday, 4. Oct. (i.e. 1 hour lesson + 1 hour exercises).
NB ! There will be no (!) lessons on Wednesday, 19. Sept. and on Thursday, 20. Sept. !
The next lesson is supposed to be on Wednesday, 26. Sept. !
Students who are interested in actuarial mathematics are encouraged to follow the invitation of the Actuarial Society of South Africa to participate in the 2019 IAA Colloquium. See http://www.colloquium2019.org.za/ for more information about this event.
lecture notes: Part1, Part2, Part3, Part4, Part5
exercises: Exercises1, Exercises2, Exercises3, Exercises4, Exercises5, Exercises6, Exercises7, Exercises8, Exercises9
solutions: Ex1Prob5, Ex2Prob5, Ex3Prob245, Ex4Prob156, Ex5Prob256, Ex6Prob345, Ex7Prob25, Ex8Prob1356, Ex9Prob6
compendium measure/probability theory: compendium
In our last lessons (30., 31. Aug. and 5., 6. Sept.) we finished our crash course on basic notions and results from the theory of stochastic processes (see ch. 2 in the book of B. Øksendal or ch. 1-5 in in the book of Cohen, Elliott). Further, we also proved the existence of Brownian motion by using N. Wiener´s arguments from the 1920ties.
On 12. and 13. Sept. we aim at continuing with our discussion of the construction of Ito-integrals with respect to the Brownian motion (see ch. 3 in Øksendal or ch. 12 in Cohen, Elliott).
Vi starter opp med kurset på onsdag, 29. august, kl. 10:15-12:00, NHA, rom 1020 !
We start with the course on Wednesday, 29. August, 10:15-12:00, NHA, room 1020 (so there are no lessons 22./23. August) !
Course books:
1. Bernt Øksendal: Stochastic Differential Equations, 2013. Springer. (6th Edition Corrected Printing).
2. Cohen, S. N., Elliott, R. J.: Stochastic Calculus and Applications. Birkhäuser, 2nd edition (2015).
3. Chung, K.L., Williams, R. J.: Introduction to Stochastic Integration. Birkhäuser, 2nd edition (2013).
Chapters in the corresponding books:
1. Definition of stochastic integrals with respect to integrators given by right continuous local martingales (see Ch. 2, 3 in Chung, Williams or Ch. 12 in Cohen, Elliot). Discussion of stochastic integrals with respect to the Brownian motion (see Ch. 3 in Øksendal).
2. Ito-formula and martingale representation theorem with respect to the Brownian motion (see Ch. 4 in Øksendal)
3. Stochastic differential equations driven by Brownian motion (Ex...