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Stochastic Processes - MAT00018M
- Department: Mathematics
- Module co-ordinator: Prof. Zdzislaw Brzezniak
- Credit value: 10 credits
- Credit level: M
- Academic year of delivery: 2021-22
- See module specification for other years: 2022-23
Related modules
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Autumn Term 2021-22 |
| B | Spring Term 2021-22 |
Module aims
- To introduce students to a range of mathematical models that take account of the stochastic (random) fluctuations that are always present in the real world;
- To demonstrate the circumstances in which continuous-time stochastic models give results that are different to those from deterministic models that neglect the random effects;
- To provide a range of mathematical techniques and approximations that can be used to make analytic predictions from stochastic models;
Module learning outcomes
At the end of the module you should be able to...
- appreciate the uses for stochastic models, their characteristics and limitations;
- explain the concept of continuous-time stochastic processes and the Markov property;
- give examples of applications of stochastic processes;
- formulate and analyse Markov models in continuous time;
- calculate (conditional) probabilities of events and expectations of variables described by simple Markov processes like the Poisson process or the Wiener process
- determine transition rates and stationary distributions of birth-death processes
- understand the basic properties of the Wiener process;
- define the Ito stochastic integral and give its important properties.
- Apply the Ito Lemma to verify if certain processes are martingales and to find solutions of certain stochastic differential equations;
;
Module content
- Principles of stochastic modelling
- The need for models
- Stochastic vs. deterministic models.
- Continuous-time stochastic processes
- The relevance of the Markov property
- Discrete state-space: Markov jump processes
- Poisson process
- Birth-death processes
- Kolmogorov equations (master equations)
- Stationary distributions
- Continuous state-space, stochastic calculus
- Wiener process
- Stochastic differential equations and Ito’s calculus. Examples include: geometric Brownian motion;
- Stochastic integration
Coursework Assessment: Students will be required to complete directed private study with assessed coursework. Students will complete:
- A reading study on discrete-time Markov chains, with a detailed assignment for a particular application and a written report on a topical system.
Assessment
| Task | Length | % of module mark |
|---|---|---|
| Essay/coursework Coursework |
N/A | 25 |
| Online Exam -less than 24hrs (Centrally scheduled) Stochastic Processes |
2 hours | 75 |
Special assessment rules
None
Reassessment
| Task | Length | % of module mark |
|---|---|---|
| Online Exam -less than 24hrs (Centrally scheduled) Reassessment: Examination - 2hrs -Stochastic Processes |
2 hours | 100 |
Module feedback
- Online feedback on coursework
- Discussion of coursework in seminars following due date
- Typeset model solutions to exercise problems
- Examination result delivered in Week 10 of Spring Term, with model solutions and examiner’s comments available earlier.
Indicative reading
- G R Grimmett & D R Stirzaker, Probability and random processes, OUP.
- C W Gardiner, Handbook of stochastic methods, Springer.
- T Mikosch, Elementary stochastic calculus with finance in view, World Scientific.
- Z Brzezniak & T Zastawniak, Basic Stochastic Processes, Springer.
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