Undergraduate Programme and Module Handbook 2010-2011 (archived)

# Module MATH4091: STOCHASTIC PROCESSES IV

## Department: Mathematical Sciences

### MATH4091: STOCHASTIC PROCESSES IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2011/12 and alternate years thereafter | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Linear Algebra II (MATH2021) AND Analysis in Many Variables II (MATH2031) AND Probability and Geometric Topology II (MATH2151) OR Probability and Actuarial Mathematics II (MATH2161) OR Codes and Probability II (MATH2571) OR Contours and Probability II (MATH2561))

#### Corequisites

- None.

#### Excluded Combination of Modules

- Stochastic Processes III (MATH3251).

#### Aims

- This module continues on from the treatment of probability in MATH1012 (Core Mathematics A).
- It is designed to introduce mathematics students to the wide variety of models of systems in which sequences of events are governed by probabilistic laws.
- Students completing this course should be equipped to read for themselves much of the vast literature on applications to problems in physics, engineering, chemistry, biology, medicine, psychology and many other fields.

#### Content

- Conditional probability and sigma fields.
- Examples of Markov chains.
- Discrete parameter Martingales and renewal theory.
- General renewal theory
- Poisson processes.
- Continuous time Markov chains.
- Topics chosen from: stationary Gaussian processes, Brownian motion, percolation theory, contact process.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve novel and/or complex problems in Stochastic Processes.
- have a systematic and coherent understanding of theoretical mathematics in the field of Stochastic Processes.
- have acquired coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Probability.
- Discrete Parameter Martingales
- Renewal Theory.
- Poisson processes.
- Continuous time Markov processes.

Subject-specific Skills:

- In addition students will have highly specialised and advanced mathematical skills in the following areas: Modelling, Computation.

Key Skills:

- Students will be able to study independently to further their knowledge of an advanced topic.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
- Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
- Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems. The Subject material assigned for independent study will form part of the examined material.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 40 | 2 per week | 1 Hour | 40 | |

Preparation and Reading | 160 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

three-hour examination | 100% |

#### Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University