Tentative lecture plan
This plan is tentative and may be changed.
BØ = Bernt Øksendal’s book
| Week | Topics | Textbook and notes | Comments |
|---|---|---|---|
| 34/35 | Introduction to measure and integration Sigma algebras, measures, mesurable functions, More on mesurable functions, the Lebesgue integral. | Note on Measure and Integration See also Øksendal Appendix B | |
| 36 | Big theorems for the Lebesgue integral, Lp-spaces, Review of probability theory Probability space, random variables, distribution. Moments, characteristic functions, multivariate Gaussian variables. | 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 | Independence, Borel-Cantelli lemmas. Conditional Expectation. | Note on Measure and Integration | More details in the book by Jacod and Protter. |
| 38 | Stochastic processes Definition, finite dimensional distributions Brownian motion definition, distributional properties. Invariance and sample path properties of BM, further remarks, motivation, Levy construction | Note on Brownian motion Øksendal chp. 2.2 Evans chp 2.I and 3 | More details in the books by Schilling and Evans. |
| 39 | Levy construction of BM - the proof. Martingales definition (discrete and continuous), filtration, adapted processes, examples, inequality, closedness in L2, martingale transform. | Note on Brownian motion, Øksendal p 31, 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 | Stochastic integrals White noise and SDEs Stieltjes integrals Elementary functions and Ito isometry The Itô integral in \(L^2\), definition. Properties, examples, martingale, continuity a.s., generalizations. | Øksendal chp. 3, Note on Martingales and the Itô Integral | Read yourselves: A comparision of Ito and Stratonovich integrals, Øksendal p. 35-37 |
| 41 | Itô processes Itô's formula with proof, mean square continuous processes. Multi-dimensional versions Stochastic Differential Equations Strong solutions Example | Øksendal chp. 4.1-4.2, 5.1 Note on Mean Square Continuous Processes and Proof of Ito's formula | Read yourselves: Øksendal chp. 4.3 (proofs not relevant for the exam) |
| 42 | Existence and uniqueness of strong solutions Weak solutions, definition and uniqueness | Øksendal chp. 5 Evans Lecture Note chp. 5.A (the definition) | Read yourselves: Øksendal chp. 5.3 (discussion on weak solutions and uniqueness) |
| 43 | Project work | No lectures, office hours during lecture times |
|
| 44 | Project work | No lectures, office hours during lecture times |
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| 45 | 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 |
| 46 | 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 |
| 47 | 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. |