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Stochastic Processes - MAT00030H

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• Department: Mathematics
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2022-23

Related modules

• None

Prohibited combinations

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

Module will run

Occurrence Teaching period
A Autumn Term 2022-23

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

Indicative assessment

Closed/in-person Exam (Centrally scheduled) 100

None

Indicative reassessment

Closed/in-person Exam (Centrally scheduled) 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.