Lecture plan
This plan is tentative and may be changed.
BØ = Bernt Øksendal’s book
Week | Topics | Textbook and notes | Comments |
---|---|---|---|
35 | Introduction to measure and integration Sigma algebras, measures, mesurable functions, the Lebesgue integral, main theorems Lp-spaces | Note on Measure and Integration See also Øksendal Appendix B | |
36 | Review of probability theory Probability space, stochastic variables, distribution, moments, characteristic functions, multivariate Gaussian variables Independence. Borel-Cantelli lemmas Conditional Expectation | Note on Measure and Integration Øksendal Appendix A and B See also Øksendal sec. 2.1 | |
37 | Conditional Expectation Stochastic processes Definition Brownian motion motivation, definition, properties finite dimensional distributions | Note on Measure and Integration Note on Brownian motion Øksendal chp 2.1 Evans Lecture note chp 3.A, 3.B | |
38 | Brownian motion - further remarks, sample path properties, Levy-Ciesielski construction Martingales definition (discrete and continuous), filtration, adapted processes, examples, inequality, closedness in L2, martingale transform | Note on Brownian motion Øksendal chp. 2.1 + page 31 Evans chp 2.I and 3 | |
39 | Stochastic integrals White noise and SDEs Stieltjes integrals Elementary functions and Ito isometry The Itô integral Definition and properties, martingale, continuity a.s. | Øksendal chp. 3 Note on Martingales and the Itô Integral | Read yourselves: A comparision of Ito and Stratonovich integrals, Øksendal p. 35-37 |
40 | Itô processes Itô's formula with proof Multi-dimensional versions Stochastic Differential Equations Strong solutions Example | Øksendal chp. 4.1-4.2, 5.1 Evans Lecture note chp 4.D, 4.E (for proofs of Ito's formula) | Read yourselves: Øksendal chp. 4.3 (proofs not relevant for the exam) |
41 | 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) |
42 | Solutions to some linear SDEs Pysical Browninan motion Diffusion processes Introduction, Markov processes, Itô Diffusions | Øksendal chp. 5 Note on Linear SDEs and Physical Brownian Motion Note on Informal Comments on Itô Diffusions | Espen is in Spain Harald Krogstad will give the lectures |
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 | Markov processes, Itô Diffusions Stopping times, Dynkin’s Formula, the generator, The Dirichlet problems for elliptic PDEs Brownian motion in Rn | Øksendal chp. 7 Note on Informal Comments on Ito Diffusions | |
46 | Brownian motion in Rn hitting and exit times and probabilities Kolmogorov's backward equation | Øksendal chp. 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. |