Lecture plan
This plan is tentative, it will be updated continuously.
Because of travelling, there will be more lectures some weeks and no lectures others.
BØ = Bernt Øksendal’s book
| Week | Topics | Textbook and notes | Comments |
|---|---|---|---|
| 34 | Introduction to the course. Introduction to measure and integration Sigma algebras, measures, measurable functions. | Note on Measure and Integration Theory by Krogstad. See also Øksendal Chp 2 and Appendix B | |
| 35 | More on mesurable functions, the Lebesgue integral. Big theorems for the Lebesgue integral, Lp-spaces. Review of probability theory Probability space, random variables, distribution. | Note on Measure and Integration See also Øksendal Appendix B | More details in the book by Jacod and Protter. |
| 36 | Moments, characteristic functions, multivariate Gaussian variables. Independence, Borel-Cantelli lemmas. | Note on Measure and Integration Øksendal Appendix A and B \\See also Øksendal sec. 2.1 | More details in the book by Jacod and Protter. |
| 37 | Conditional Expectation. Stochastic processes Definition, finite dimensional distributions Brownian motion (BM) definition, distributional properties. | Note on Measure and Integration \\ Note on Brownian motion Øksendal chp. 2.2 Evans chp 2.I and 3 | More details in the book by Jacod and Protter. |
| 38 | Invariance and sample path properties of BM, further remarks, motivation, Levy construction Levy construction of BM - the proof. | Note on Brownian motion Øksendal chp. 2.2 Evans chp 2.I and 3 | More details in the books by Schilling & Partzsch and Evans. |
| 39 | Martingales definition (discrete and continuous), filtration, adapted processes, examples, inequality, closedness in L2, martingale transform. Stochastic integrals White noise and SDEs Stieltjes integrals Elementary functions and Ito isometry The Itô integral in \(L^2\), definition. | Note on Brownian motion, Øksendal p 31, Note on Martingales and the Itô Integral Øksendal chp. 3, Note on Martingales and the Itô Integral | The proof of the Levy construction presented in class is based on the note of Krogstad, but we have filled in missing parts. A nice and fairly complete proof can be found in the book by Schilling. |
| 40 | Properties, examples, martingale, continuity a.s., generalizations. Itô processes Itô's formula with proof, mean square continuous processes. Proof of Ito's formula. Multi-dimensional versions | Øksendal chp. 3, Note on Martingales and the Itô Integral Øksendal chp. 4.1, 4.2 Note on Mean Square Continuous Processes and Proof of Ito's formula | Extra lecture this week (3 lectures). Read yourselves: A comparision of Ito and Stratonovich integrals, Øksendal p. 35-37 Read yourselves: Øksendal chp. 4.3 (proofs not relevant for the exam) |
| 41 | Stochastic Differential Equations Strong solutions. Example. Stability and uniqueness of strong solutions. Proof of uniqueness. Existence of strong solutions. | Øksendal chp. 5 Evans Lecture Note chp. 5.A (the definition), see also the books by Kuo and Schilling. | OBS: There seems to be a mistake in the proof of existence for SDEs in the 6th edition of Øksendal. The proof in previous editions is correct but longer. We will use the shorter proof given in Schilling, see my handwritten notes: Strong solution of SDEs. Uniqueness and Existence Read yourselves: Øksendal chp. 5.3 (discussion on weak solutions and uniqueness) This part was discussed only orally in class. |
| 42 | Project work. Espen in Madrid. | No lectures. | |
| 43 | Project work | No lectures, office hours during lecture times, see info page. |
|
| 44 | Solutions to some linear SDEs, integrating factor Diffusion processes Introduction, Markov processes, Itô Diffusions Stopping times, Dynkin’s Formula, the generator, The Dirichlet problems for elliptic PDEs | Øksendal chp. 5.1 and 7.1-7.4 Note on Linear SDEs and Physical Brownian Motion Note on Informal Comments on Itô Diffusions | Read yourselves: Note on Linear SDEs and Physical Brownian Motion |
| 45 | Brownian motion in Rn, hitting and exit times and probabilities, Kolmogorov's backward equation | Øksendal chp. 7.4 and 8.1 Note on Fokker-Planck eqns. and Kolmogorov forward and backward eqns. | Read yourselves: Proof of Theorem 8.1.1 b) in Øksendal |
| 46 | The Fokker-Planck or Kolmogorov's Forward Equations, Øksendal problem 8:8.2 The Resolvent Feynman-Kac Formula, Itô diffusions and Martingales | Øksendal chp. 8.2-8.3 Note on Fokker-Planck eqns. and Kolmogorov forward and backward eqns. |