MA50251: Applied SDEs: SAMBa/TCC Graduate Course 2017/18

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Problem Sheets

iPython Notebooks

In the first half of the course, some examples will be given using iPython. The notebooks (and a complete .pdf) are made available below. To get started with iPython, download the software from here: iPython. A dictionary for Matlab users can be found here.

This graduate course will look at Stochastic Differential Equations from an Applied perspective. In particular, we will not assume a deep probabilistic background, and the emphasis will tend to be on the applications, although hopefully there will also be something to interest students with a more classical probability background.

The course will roughly break into two parts: after some motivation and discussion of introductory problems, we will review much of the background theory: in particular, we will provide an overview of stochastic integration in a Brownian filtration, and some SDE theory and key results, following the presentation in Øksendal's book.

Problem sheets will be set weekly, and (for those students who need to be assessed), there will be coursework (25%) and a final exam (75%).

The course is timetabled at 9:15-11.05 on Monday in 4W 1.7. A full timetable of planned lectures is here, see also content below.

Lecture 1 (12/2/18, AC)
Introduction, Motivating discussion, Brownian motion, Donsker’s invariance principle, Quadratic variation.
Lecture 2 (19/2/18, AC)
Stochastic Integration: Construction, Properties, Itô Isometry, Stratonovich integral.
Lecture 3 (26/2/18, AC)
Stochastic Calculus: Itô’s Lemma, Integration by Parts.
Lecture 4 (5/3/18, AC)
Stochastic Differential Equations: Existence and Uniqueness, weak and strong solutions. Diffusion Processes: Markov Property, Generators, Boundary Value Problems.
Lecture 5 (12/3/18, TS)
Numerical methods for stochastic differential equations: Euler–Maruyama and Milstein methods, Modes and rates of convergence, Experiments.
Lecture 6 (9/4/18, TS)
Fokker–Planck equation. Derivation and example solutions. Ergodicity and invariance measures. Brownian dynamics and Langevin equations.
Lecture 7 (16/4/18, TS)
Exit-time problems. Formulation of PDE for mean-exit time. Small-noise limits and Kramers’ rate. Metastability.
Lecture 8 (23/4/18, TS)
Parameter estimation. Lamperti transformation. Estimating diffusion coefficient. Derivation by Girsanov. Minimum likelihood estimation. Examples.

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Last modified: 12/2/18
Maintainer: Alex Cox