Applied SDEs
This graduate course looks at Stochastic Differential Equations from an applied perspective. In particular, we do not assume a deep probabilistic background, and the emphasis tends to be on the applications, although hopefully there is also something to interest students with a more classical probability background.
The course breaks roughly into two parts: after some motivation and discussion of introductory problems, we review much of the background theory — in particular an overview of stochastic integration in a Brownian filtration, and some SDE theory and key results, following the presentation in Øksendal’s book.
Lectured by Alex Cox (AC) and Tony Shardlow (TS).
Problem sheets are set weekly, and (for students who need to be assessed) there is coursework (25%) and a final exam (75%). The course is timetabled at 9:15–11:05 on Mondays in 4W 1.7. A full timetable of planned lectures is here; see also the syllabus below.
Lecture notes
Problem sheets
- Sheet 1 · questions · solutions
- Sheet 2 · questions · solutions
- Sheet 3 · questions · solutions
- Sheet 4 · questions · solutions
- Sheet 5 · questions · solutions
- Sheet 6 · questions · solutions
- Sheet 7 · questions · solutions
- Sheet 8 · questions · solutions
Notebooks
In the first half of the course, some examples are given using Python. The notebooks (and a complete pdf of each) are below. To get started, install Jupyter/IPython; a dictionary for MATLAB users is here.
- Stochastic Integration · notebook · pdf
- Quadratic Variation · notebook · pdf
- Itô vs Stratonovich · notebook · pdf
- Simple Integrands · notebook · pdf
- Solving PDEs via Monte Carlo · notebook · pdf
Syllabus
- 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 invariant measures. Brownian dynamics and Langevin equations.
- Lecture 7 (16/4/18, TS)
- Exit-time problems. Formulation of the PDE for mean exit time. Small-noise limits and Kramers’ rate. Metastability.
- Lecture 8 (23/4/18, TS)
- Parameter estimation. Lamperti transformation. Estimating the diffusion coefficient. Derivation by Girsanov. Maximum likelihood estimation. Examples.