BSc Mathematics with Placement Year

A stochastic process is a collection of random variables that describe the progression of a non-deterministic system as a function of time.

Stochastic processes are used as modelling tools within a wide range of applications arising virtually in any area of modern science and engineering where randomness plays a role.

The unit aims to present the most important classes of stochastic processes by focusing on the fundamental examples, describing the meaning of their sample paths, and explaining their role in modelling specific science and/or engineering phenomena.

These classes of stochastic processes are of fundamental interest in (i) Mathematical Finance, (ii) Actuarial Science, and (iii) Statistics, in addition to being of interest in themselves as fundamental entities of modern (iv) Probability Theory that provide fascinating connections to (v) Mathematical Analysis.

Syllabus: (with approximate times)
1. Introduction (stochastic process, sample path, increment, marginal
    law, first entry time) [1 week]
2. Random walk (definition, basic properties, marginal law, examples
    of application) [3 weeks]
3. Poisson process (definition, basic properties, marginal law, examples
    of application) [3 weeks] 
4. Wiener process [Brownian motion] (definition, basic properties, marginal
    law, examples of application) [3 weeks] 
5. Stationary process (definition, covariance function, examples of
    application) [1 week] 
 

MATH27712 Co-Requisite: Students must be enrolled on MATH27720 in order to enrol onto MATH27712

The unit aims to:

- Present the most important classes of Stochastic Processes by focusing on the fundamental examples, describing the meaning of their sample paths, and explaining their role in modelling specific science and/or engineering phenomena;

- Provide an overview of Stochastic Processes and explain what the students can expect in related directions from the probability-based units in year 3, 4 and beyond.
 

• Define a stochastic process and describe its meaning as a modelling tool of a specific science/engineering phenomenon.
• Describe the structure of the sample paths of a stochastic process and derive their basic properties.
• Derive the marginal law of a stochastic process, calculate its expectation/variance, and study its asymptotic behaviour.
• Define the first entry time of a stochastic process and apply the derived results in a variety of applied settings.

Summer exam    
2 hours    General feedback provided after exam is marked.
    

Stochastic processes    Bass, Richard F., author.    Cambridge University Press    2011    ISBN: 9780511997044
Stochastic processes : an introduction    Jones, P. W.    CRC Press Taylor & Francis Group    2018    ISBN: 9781498778114
A first course in stochastic processes    Karlin, Samuel, 1924-2007, author.    Academic Press    1975    ISBN: 9780080570419

Tuition fees for home students commencing their studies in September 2027 will be £10,050 per annum. Tuition fees for international students will be £38,100 per annum.

For general information please see the undergraduate finance pages.

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).