Module: Stochastic Modelling

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Stochastic Modelling
Module Code	MTH1050 (ITS: MS308)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	8	Credit Rating	7.5

Description

To give a comprehensive introduction to Markov chains, Markov jump processes and their application to actuarial science.

Learning Outcomes

1. Construct Markov chain models for actuarial and financial processes.
2. Analyse any given chain in a systematic way, including determining its asymptotic behaviour.
3. Prove the main theorems governing Markov chains in discrete and continuous time.
4. State the definitions of the main concepts underlying the theory of Markov chains and demonstrate an understanding of these through examples and counter-examples.

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	36	No Description
Tutorial	12	No Description
Laboratory	15	Practising computations and simulations using R.
Independent Study	135	No Description
Total Workload: 198

Section Breakdown
CRN	11216	Part of Term	Semester 1
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC1	Best Mark	N
Module Co-ordinator	Martin Venker	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
In Class Test	Lab test using R.	25%	Week 10
Formal Examination	End-of-Semester Final Examination	75%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Stochastic Modelling
Review of basic probabilistic concepts, the various types of stochastic processes, stationarity, Markov processes, the Chapman-Kolmogorov equations, stationary probability distributions. [CS2 - 3.1]

Markov Chains
Solution of the Chapman-Kolmogorov equation in matrix form, transition graph, finding the stationary distribution, actuarial examples; two-state chains; the limiting distribution of finite Markov chains, irreducibility and aperiodicity, exponential convergence; infinite Markov chains, criteria for recurrence, the limiting distribution and its relation to mean recurrence times; applications: queues, random walks with various boundary conditions. [CS2 - 3.2]

Markov Jump Processes
The infinitesimal generator, the forward and backward equations, solution in exponential form; holding times, exponential distribution, jump chain; the limiting distribution of a finite Markov jump process and its connection to mean recurrence times; the case of infinite state spaces, the integral form of the backward equation, the minimal process, conservative processes; the Poisson process and actuarial models; inhomogeneous Markov jump processes, time-dependent transition rates, the backward equation in differential and integral forms, residual holding times. [CS2 - 3.3].

Indicative Reading List

Books:

Bhattacharya, R.N., and Waymire R.C: 1990, Stochastic Processes with Applications, Wiley, NewYork,
Grimmett, G.R. and Stirzaker, D.R.: 1992, Probability and Random Processes, 2-nd, Oxford UP, Oxford,
Norris, JR: 1997, Markov Chains, Cambridge UP, Cambridge,
A.N. Other: 0, Acted material for CT4 subject , models ,

Articles:
None

Other Resources

None