MAGIC: Stochastic Processes (MAGIC065)

The two parts of the course - see the syllabus - will be presented in parallel: part 1 on Friday and part 2 on Monday. (In particular, we start on Monday 10.10.2011 with the first lecture from part 2.) There is no stone wall between the two parts. Some mild cross-references will remain, however, the idea is that each part may be attended (and assessed) independently, although, naturally, we encourage the audience to attend both parts. The team PS. Please, do not forget to register on each lecture you attend.

It is desirable to know something about Markov chains.

======================= Fridays Lythe (weeks 1-4) Voss (weeks 5-8) Molina-Paris (weeks 9-10) ======================= 1-1 Gambler's ruin. Discrete random variables. Continuous random variables. 1-2 Random walk, discrete-time Markov chains. 1-3 Branching processes. Continuous-time Markov Chains. Birth and death processes. Gillespie algorithm. 1-4 Stationary distributions, quasi-limiting distributions. 1-5 Stochastic processes. Wiener process. Diffusion equation. 1-6 The reflection principle and passage times. Conditional hitting probability. 1-9 Applications to immunology. ====================== Mondays (Veretennikov) ====================== 2-1 Stochastic processes; some measure theory; Kolmogorov continuity theorem. 2-2 Filtrations and conditional expectations. 2-3 Wiener measure. 2-4 Stochastic Ito integrals. 2-5 Stopping times; martingales; Kolmogorov and Doob theorems. 2-6 Ito formula. 2-7 Stochastic differential equations, existence and uniqueness of solutions. 2-8 Passage times, links to Laplace and Poisson equations; Dynkin and Feynman-Kac formulae. 2-9 Girsanov change of measure; weak solutions of SDEs. 2-10 Dependence of solutions of SDEs from initial data; Markov property of solutions.

1. There will be one assignment consisting of two parts - for two parts of the module - and each of which will include five of six questions. 2. It is allowed to take an exam for one part of the course, Friday or Monday, or both. (For each part you would get a half of the total amount of credits.) To sit either part, you have to choose four questions from this part. I.e., to sit the first part, you choose four questions from the first part and to sit both parts you choose four questions FROM EACH PART, altogether EIGHT. 3. The assignment will be available from Friday 16.12.2011. Your reports are due by 16.01.2012. We then intend to return your marks (at pass/fail scale as suggested by Magic) by 01.02.2012. 4. To pass this exam successfully, you have to satisfy the requirement in item 2 above and solve reasonably well a bit more than half of all chosen questions, that is, more than 2/4 for one part and 4/8 for both parts; it roughly corresponds to 60