Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title APPLIED STOCHASTIC MODELS
Code MATH360
Coordinator Dr GH Berzunza Ojeda
Mathematical Sciences
Gabriel.Berzunza-Ojeda@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2024-25 Level 6 FHEQ First Semester 15

Aims

To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for 'dynamic' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.

Learning Outcomes

(LO1) Use the theory of conditional probability to calculate and analyse the likelihood of an event occurring, given the occurrence of a previous event or outcome.

(LO2) Construct and analyse continuous-time Markov chains, including proving their key properties.

(LO3) Prove several key properties of Brownian motion and similar processes.

(LO4) Apply the theory of continuous-time Markov chains and Brownian motion to model or solve real-world problems in epidemiology, mathematical biology, financial mathematics, and other fields.

(S1) Problem solving skills

(S2) Numeracy

Syllabus
 

·Introduction and preliminaries: Conditional Probability/expectation with a continuous random variable, Conditional densities, Random processes; Continuous-time stochastic processes.
·Basic concepts of continuous-time Markov chains:  Definitions, Transition probabilities and their properties, Chapman-Kolmogorov equations, Absolute distributions; Transition rate matrix, Kolmogorov forward and backward equations.
·Properties of continuous-time Markov chains:  Classification of states, Absorbing state and probabilities, Recurrence and transience, Ergodicity, Limiting and stationary probabilities.
·Application and examples of continuous-time Markov chains: Finite-state chains, A selection of examples including some of the following: Pure birth process, Pure death process, Birth-and-death processes; M/M/1 queue and other Markovian queuing models, Simple branching processes; Application to epidemiology, Predator-prey processes, Competition proc esses; Population processes.
·Brownian motion processes: Multi-normal distributions, Standard BM, BMs with drift and their properties, Absolute distribution, Transition densities, Kolmogorov forward and backward equations, Co variance functions.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH254 STATISTICS AND PROBABILITY II 2023-24 

Co-requisite modules:
MATH362 APPLIED PROBABILITY 2023-24 

Modules for which this module is a pre-requisite:
 

Programme(s) (including Year of Study) to which this module is available on a required basis:
 

Programme(s) (including Year of Study) to which this module is available on an optional basis:
 

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
written exam  120    70       
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Class Test  60    30