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 Y Zhang
Mathematical Sciences
Yi.Zhang@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ Second 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

After completing the module students should have a grounding in the theory of continuous-time Markov chains and diffusion processes. They should be able to solve corresponding problems arising in epidemiology, mathematical biology, financial mathematics, etc.


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 proce ss, Birth-and-death processes; M/M/1 queue and other Markovian queuing models, Simple branching processes; Application to epidemiology, Predator-prey processes, Competition processes; 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, Covariance functions. 

·       Applications of Brownian motion processes: Geometric Brownian motions, Application to financial mathematics, Ito Processes, Ito’s Lemma, The Black-Scholes option pricing formula. 


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH264; MATH101; MATH103; MATH162  

Co-requisite modules:

 

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:

Programme:G1R9 Year:4 Programme:G100 Year:3 Programme:G101 Year:3/4 Programme:G110 Year:3 Programme:G1N3 Year:3 Programme:NG31 Year 3 Programme:GG13 Year:3 Programme:GL11 Year:3 Programme:GR11 Year:4 Programme:GV15 Year:3 Programme:MMAS Year:1

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Unseen Written Exam  2.5 hours  Second semester  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) Full marks can be obtained for answers for FIVE questions. Only the best FIVE answers will be counted.  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes