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# Stochastic Processes - MAT00030H

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• Department: Mathematics
• Module co-ordinator: Prof. Zdzislaw Brzezniak
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2021-22

## Related modules

• None

### Prohibited combinations

Pre-requisites for Natural Sciences students: must have taken Statistics Option MAT00033I.

## Module will run

Occurrence Teaching cycle
A Autumn Term 2021-22

## Module aims

• To introduce students to a range of mathematical models that take account of the stochastic (random) fluctuations that are always present in the real world;
• To demonstrate the circumstances in which continuous-time stochastic models give results that are different to those from deterministic models that neglect the random effects;
• To provide a range of mathematical techniques and approximations that can be used to make analytic predictions from stochastic models;

## Module learning outcomes

At the end of the module the student should be able to:

• appreciate the uses for stochastic models, their characteristics and limitations;

• explain the concept of continuous-time stochastic processes and the Markov property;

• give examples of applications of stochastic processes;

• formulate and analyse Markov models in continuous time;

• calculate (conditional) probabilities of events and expectations of variables described by simple Markov processes like the Poisson process or the Wiener process;

• determine transition rates stationary distributions of birth-death processes;

• discuss the properties of the Wiener process;

• define the Ito stochastic integral and give its important properties;

• apply Ito's Lemma to find solutions of certain stochastic differential equations;

## Module content

• Principles of stochastic modelling

• The need for models

• Stochastic vs. deterministic models.

• Continuous-time stochastic processes

• The relevance of the Markov property

• Discrete state-space: Markov jump processes

• Poisson process

• Birth-death processes

• Kolmogorov equations (master equations)

• Stationary distributions

• Continuous state-space, stochastic calculus

• Wiener process

• Stochastic differential equations and Ito’s calculus. Examples include: geometric Brownian motion; Ornstein-Uhlenbeck process

• Stochastic integration

## Assessment

Task Length % of module mark
Online Exam
Stochastic Processes
N/A 100

None

### Reassessment

Task Length % of module mark
Online Exam
Stochastic Processes
N/A 100

## Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.