Tentative lecture plan
This plan is tentative and may be changed.
BØ = Bernt Øksendal’s book
| Week | Topics | Textbook and notes | Comments |
|---|---|---|---|
| 34 | Introduction to the course. | ||
| 35 | Introduction to measure and integration Sigma algebras, measures, measurable functions. | Chapter 2 in BØ (extremely compact). Note on Measure and Integration Theory by Krogstad. | |
| 36 | The Lebesgue integral Monotone convergence theorem, Fatou's lemma, the dominated convergence theorem Lp-spaces Review of probability theory Probability space, random variables, expectation values, independence, conditional expectation | Note on Measure and Integration Theory by Krogstad. BØ, appendix B (conditional expectation) | More details can be found in Jacod & Protter. See e.g. chapter 9 on integration theory (proofs) |
| 37 | Stochastic processes Definition. Brownian motion Definition, properties (scale and translation invariant, nowhere differentiable, quadratic variation) Levy construction. Borel-Cantelli's lemma | Note on Brownian Motion by Krogstad, BØ chp. 2.2. Kuo 2.1-2.2, 3.2, 3.4 | Even more details can be found in Schilling and Partzsch, and Evans. |
| 38 | Stochastic integrals Filtered probability spaces, F(t)-adapted processes, martingales Construction of the Ito integral Properties of the integral (Thm. 3.2.1) Ito isometry (Cor.3.1.7) | BØ 3.1-3.2 up to theorem 3.2.4 (which is left for next week) | More details can be found in Kuo, sec.4. |
| 39 | Properties of the Ito integral. Extensions of the Ito integral. The Ito formula, and how to use it. | BØ 3.2-3.3 BØ 4.1-4.2 | |
| 40 | Stochastic differential equations Introduction, some examples. Solution of linear SDEs and some properties of those. | BØ 5.1 Note on linear SDEs | |
| 41 | Existence and uniqueness results for SDE | BØ Chapter 5.2 from the 5th edition. Some notes from the Thursday lecture. | |
| 42 | Numerical solution of SDEs Euler-Maruyama and the Milstein method Error and convergence concepts Wagner-Platen series | Note on numerical SDE, part 1 Jupyter notebook Note on numerical SDE, part 2 | Start project work |
| 43 | Milstein's theorem on mean square convergence, proof included. Linear stability for SDEs. | Note on numerical SDE, part 3 | For a complete proof of the convergence theorem, see Milstein and Tretyakov, chapter 1. For more on linear stability, see e.g. this paper by Des Higham. |
| 44 | No lectures | ||
| 45 | Diffusion processes Markov processes, probability transition functions, Chapman-Kolmogorov's equation, Ito diffusion, infinitesimal generators Kolmogorov's equations. | BØ chapter 7.1, 7.3 and 8.1 Krogstad' s notes on Informal comments on Ito diffusions and Kolmogorov and Fokker-Planck equations. | |
| 46 | Stopping times, the Dunkin formula, | BØ chapter 7.2, 7.3 Evans, section 6, example 1 and 2. | |
| 47 | Summary, exam preparations (on Friday). | No lecture 25.11. |