Registry

Module Specifications

Archived Version 2014 - 2015

Module Title	Stochastic Modelling
Module Code	MS308
School	School of Mathematical Sciences
Online Module Resources
NFQ level	8	Credit Rating	7.5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

This module provides a comprehensive introduction to Stochastic Processes and their application to Actuarial Science. It does so by blending the development of the theory of Markov chains (both in disccrete and continuous time) with modelling examples . As a result , the module is of mixed "theory" and "know-how and skills" type .

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 equation, stationary probability distributions. [CT4 - (ii)]

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. [CT4 - (iii)]

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. [CT4 - (iv)]. Survival models, sickness and death, estimation of transition rates; finite population observed fora fixed time interval, truncated life-times, unbiased estimator of transition rate, asymptotic distribution, poisson approximation. [CT4 - (vii)]

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Type	Description	% of total	Assessment Date
Course Work Breakdown

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
Unavailable

Indicative Reading List

A.N. Other: 0, Acted material for CT4 subject , models ,
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,

Other Resources

None

Programme or List of Programmes

ACM	BSc Actuarial Mathematics
FM	BSc in Financial & Actuarial Mathematics
SMPSC	Single Module Professional Science

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