# ST403 - Brownian Motion

**Module code:**ST403**Module name:**Brownian Motion**Department:**Statistics**Credit:**15

Content and teaching | Assessment | Availability

## Module content and teaching

###### Principal aims

Brownian motion was originally the description given in physics for the random erratic movement of molecules. In 1905 Einstein made a detailed study in which he postulated certain properties should hold. In 1923 mathematical Brownian motion was born when a famous mathematician, Norbert Wiener, showed how to construct a random function W(t) giving the molecules “position” at time t which had Einstein’s properties.WHY IS IT INTERESTING? It is a beautiful mathematical object worth studying both for its own sake and because of the deep links it has with other areas of mathematics, particularly in analysis. Brownian motion is a fundamental tool for modelling processes which evolve randomly in time. It is used widely in many areas of applied maths and in the last few decades it has become essential to the study of financial maths as a model of stock prices.

###### Principal learning outcomes

WHAT WILL WE LEARN? Construction. According to Einstein - the function t ® W(t) must be continuous – the molecule never jumps; the displacement between times s and t, that is W(t) – W(s), should be independent of the past motion and its distribution should be Gaussian with mean zero and variance t – s; We will investigate methods of constructing such random functions. It turns out the Gaussian distribution is essential – it is impossible to do with any other distribution.

###### Timetabled teaching activities

3 hours per week

###### Departmental link

http://www2.warwick.ac.uk/fac/sci/statistics/courses/modules/years3-4/st403

###### Other essential notes

Also offered by Maths as MA4F7. Prerequisite(s): AT LEAST ONE OF: ST318 Probability Theory, MA359 Measure Theory.

## Module assessment

Assessment group | Assessment name | Percentage |
---|---|---|

15 CATS (Module code: ST403-15) | ||

D (Assessed/examined work) | Assignments worth 15% | 15% |

3 Hour April Examination | 85% |

## Module availability

This module is available on the following courses:

###### Core

N/A

###### Optional Core

N/A

###### Optional

- Undergraduate Mathematics and Statistics (BSc MMathStat) (G1G3) - Year 4
- Master of Advanced Study in Mathematical Sciences (G1PE) - Year 1
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G300) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G300) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30A) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30B) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30C) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (G30D) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30E) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30F) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30G) - Year 5
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 3
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 4
- Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated (G30H) - Year 5
- Undergraduate Discrete Mathematics (G4G3) - Year 4
- Postgraduate Taught Financial Mathematics (N3G1) - Year 1