Stochastic Processes - MAT00030H
- Department: Mathematics
- Credit value: 10 credits
- Credit level: H
- Academic year of delivery: 2022-23
Related modules
Additional information
Pre-requisites for Natural Sciences students: must have taken Statistics Option MAT00033I.
Module will run
Occurrence | Teaching period |
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A | Autumn Term 2022-23 |
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 the student should be able to:
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appreciate the uses for stochastic models, their characteristics and limitations;
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explain the concept of continuous-time stochastic processes and the Markov property;
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give examples of applications of stochastic processes;
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formulate and analyse Markov models in continuous time;
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calculate (conditional) probabilities of events and expectations of variables described by simple Markov processes like the Poisson process or the Wiener process;
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determine transition rates stationary distributions of birth-death processes;
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discuss the properties of the Wiener process;
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define the Ito stochastic integral and give its important properties;
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apply Ito's Lemma to find solutions of certain stochastic differential equations;
Module content
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Principles of stochastic modelling
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The need for models
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Stochastic vs. deterministic models.
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Continuous-time stochastic processes
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The relevance of the Markov property
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Discrete state-space: Markov jump processes
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Poisson process
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Birth-death processes
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Kolmogorov equations (master equations)
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Stationary distributions
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Continuous state-space, stochastic calculus
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Wiener process
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Stochastic differential equations and Ito’s calculus. Examples include: geometric Brownian motion; Ornstein-Uhlenbeck process
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Stochastic integration
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Indicative assessment
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Indicative reading
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G R Grimmett & D R Stirzaker, Probability and random processes, OUP.
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C W Gardiner, Handbook of stochastic methods, Springer.
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T Mikosch, Elementary stochastic calculus with finance in view, World Scientific.
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Z Brzezniak & T Zastawniak, Basic Stochastic Processes, Springer.