Undergraduate Programme and Module Handbook 2024-2025

# Module MATH3251: STOCHASTIC PROCESSES III

## Department: Mathematical Sciences

### MATH3251: STOCHASTIC PROCESSES III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- Analysis in Many Variables II (MATH2031) AND (Markov Chains II (MATH2707) OR Probability II (MATH2647))

#### Corequisites

- None.

#### Excluded Combination of Modules

#### Aims

- To develop models for processes evolving randomly in time, and probabilistic methods for their analysis, building on the treatment of probability at Levels 1 and 2. Students completing this course should be equipped to independently study much of the vast literature on applications of stochastic processes to problems in physics, engineering, chemistry, biology, medicine, psychology, and other fields.

#### Content

- Conditional expectation
- Branching processes
- Coupling theory
- Discrete renewal theory
- Discrete-time martingales and their applications
- Continuous time Markov chains
- Poisson processes
- Continuous time martingales

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will be able to:
- Rigorously define branching processes, couplings of random variables/stochastic processes, renewal processes, martingales and Markov chains in discrete and continuous time.
- Illustrate the above processes with examples.
- Explain the key theorems covered in the course (that govern the behaviour of these processes) and apply them to: calculate/estimate probabilities of events; calculate/estimate expectations of observed quantities; quantify similarity/difference between two stochastic processes; classify processes according to long term behaviour.
- (Re)construct proofs of main theorems .
- Apply theorems to solve unseen problems (appropriate to Level III and of similar type to those seen in the course) concerning stochastic processes.
- Model "real-world" or informally described systems that evolve in time subject to randomness, using appropriate stochastic processes.

Subject-specific Skills:

- In addition students will have enhanced mathematical skills in the following areas: modelling, computation.

Key Skills:

- Students will have basic mathematical skills in the following areas: problem solving, modelling, computation.

#### 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.
- Problems classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
- Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for summative assessment.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

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

Lectures | 42 | 2 per week in Michaelmas and Epiphany; 2 in Easter | 1 Hour | 42 | |

Problems Classes | 8 | Fortnightly in Michaelmas and Epiphany | 1 Hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

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

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

Written examination | 3 Hours | 100% |

#### Formative Assessment:

Eight assignments to be submitted.

■ 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