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# Stochastic Processes - MAT00018M

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• Department: Mathematics
• Module co-ordinator: Prof. Zdzislaw Brzezniak
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2022-23

• None

## Module will run

Occurrence Teaching cycle
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.

## Assessment

Task Length % of module mark
Essay/coursework
Coursework
N/A 25
Online Exam 24 hrs
Stochastic Processes
N/A 75

None

### Reassessment

Task Length % of module mark
Online Exam 24 hrs
Reassessment: Examination - 2hrs -Stochastic Processes
N/A 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.