Module: Stochastic Modelling

Module Specifications.

Current Academic Year 2024 - 2025

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Date posted: September 2024

Module Title	Stochastic Modelling
Module Code	MS308 (ITS) / MTH1050 (Banner)
Faculty	Science & Health	School	Mathematical Sciences
Module Co-ordinator	Martin Venker
Module Teachers	-
NFQ level	8	Credit Rating	7.5
Pre-requisite	Not Available
Co-requisite	Not Available
Compatibles	Not Available
Incompatibles	Not Available

Repeat examination
Array

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
Independent Study	150	No Description
Total Workload: 198

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].

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Type	Description	% of total	Assessment Date
Course Work Breakdown
In Class Test	n/a	25%	Week 9

Reassessment Requirement Type
Resit arrangements are explained by the following categories: Resit category 1: A resit is available for both* components of the module. Resit category 2: No resit is available for a 100% continuous assessment module. Resit category 3: No resit is available for the continuous assessment component where there is a continuous assessment and examination element.
* ‘Both’ is used in the context of the module having a Continuous Assessment/Examination split; where the module is 100% continuous assessment, there will also be a resit of the assessment
This module is category 3

Indicative Reading List

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 ,

Other Resources

None