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) | 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. |
| 43 | Project work | No lectures, office hours during lecture times |
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| 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. |