Stochastic Processes - MAT00018M

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  • Department: Mathematics
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23

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Module will run

Occurrence Teaching period
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 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.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 75
Essay/coursework 10
Essay/coursework 15

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 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.