Undergraduate Programme and Module Handbook 2024-2025

# Module MATH4261: Stochastic Analysis IV

## Department: Mathematical Sciences

### MATH4261: Stochastic Analysis IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Analysis III (MATH3011) Probability II (MATH2647)

#### Corequisites

- None

#### Excluded Combination of Modules

- None

#### Aims

- To introduce key concepts in conditional expectations, martingale and stochastic calculus and to explore its connection with other areas such as partial differential equations (PDEs).

#### Content

- Crash review of probability spaces and measures, theory of integrals, convergence Theo-rems; convergences of sequences of measurable functions;
- Radon-Nikodym Theorem; Conditional expectations;
- Martingales, submartingales and filtrations, submartingale inequality, upcrossings and down-crossings inequalities, submartingale convergence theorem, stopping and optional times and the Optional Sampling Theorem, Doob-Meyer Decompositions and quadratic variation pro-cesses;
- Constructions of Brownian motions, finite dimensional distributions, Kolmogorov’s Con-sistency Theorem, Kolmogorov’s Continuity Theorem, weak convergence, tightness and the Wiener measure, Levy’s modulus of continuity of Brownian motions; • Constructions of stochastic integrals, local martingales and localizations, Ito's formula, Girsanov Theorem, Burkholder-Davis-Gundy inequality;
- Stochastic differential equations, existence and uniqueness of strong solutions, Markov property, strong Markov property, Markovian semigroups and infinitesimal generators, Feynman-Kac formula, Fokker-Planck operators and forwards and backwards Kolmogorov equations, existence and estimates of probability density;
- Malliavin calculus: an introduction.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will:
- Be able to solve novel and/or complex problems in the field of Stochastic Analysis.
- Have an understanding of specialised and complex theoretical mathematics in the field of Stochastic Analysis.
- Have mastered a coherent body of knowledge of these subjects demonstrated in the following topic areas: measure and integrations; various convergence theorems of integrations and notions of variety of different convergence of sequence of measurable functions; conditional expectations and martingales; Brownian motions, stochastic integrals and stochastic differential equations; Markov property, Chapman-Kolmogorov equations and Fokker-Planck equations.
- Be able to reproduce theoretical mathematics related to this course at a level appropriate for Level 4, including key definitions and theorems.

Subject-specific Skills:

- Students will have developed advanced technical and scholastic skills in the area of Stochastic Analysis.

Key Skills:

- Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

#### 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.
- Problem 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 the 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 |

End of year written examinations | 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