BSc Mathematics with Finance

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] 
 

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.
    

[1] Bass, R. (2011). Stochastic Processes. Cambridge Univ. Press, (390 pp).
[2] Doob, J. L. (1953). Stochastic Processes. John Wiley & Sons, (654 pp).
[3] Jones, P. W. & Smith, P. (2018). Stochastic Processes: An Introduction. Chapman & Hall, (255 pp).
[4] Karlin, S. & Taylor, H. M. (1975). A First Course in Stochastic Processes. Academic Press, (557 pp).