|
Module Title |
Stochastic Modelling
|
|
Module Code |
MS308
|
|
School |
School of Mathematical Sciences
|
Online Module Resources
|
|
Level |
3
|
Credit Rating |
7.5
|
|
Pre-requisite |
None
|
|
Co-requisite |
None
|
|
|
Module Aims
|
To give a comprehensive introduction to the Markov Chains, Markov Jump Processes and their application to Actuarial Science.
|
|
Learning Outcomes
|
As a result of this module the students will have a good understanding of the most important properties of Markov chains and Markov jump processes. They will gain experience of these as tools for modelling in Actuarial Science.
|
|
Indicative Time Allowances
|
|
|
Hours
|
|
Lectures |
36
|
|
Tutorials |
12
|
|
Laboratories |
0
|
|
Seminars |
0
|
|
Independent Learning Time |
64.5
|
|
|
|
|
Total |
112.5
|
|
Placements |
|
|
Assignments |
|
|
|
NOTE
|
Assume that a 7.5 credit module load represents approximately 112.5 hours' work, which includes all teaching, in-course assignments, laboratory work or other specialised training and an estimated private learning time associated with the module.
|
|
Indicative Syllabus
|
- 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, asymptomatic distribution, poisson approximation. [CT4 - (vii)]
|
| Assessment | | Continuous Assessment | 25% | Examination Weight | 75% |
|
|
Indicative Reading List
|
- Acted material for CT4 subject ‘models’.
- Bhattacharya, R.N., and Waymire R.C., Stochastic Processes with Applications, New-York, Wiley, 1990.
- Grimmett, G.R. and Stirzaker, D.R., Probability and Random Processes, 2dn ed. Oxford University Press, 1992.
- Karlin, S. and Taylor, H.M., A First course in Stochastic Processes, 2nd ed., New York Academic Press, 1975.
- Norris, J.R., Markov Chains, Cambridge University Press, 1997.
|
|
|
|
Programme or List of Programmes
|
| BQF | BSc in Quantitative Finance |
| BSSA | Study Abroad (DCU Business School) |
| BSSAO | Study Abroad (DCU Business School) |
| ECSA | Study Abroad (Engineering & Computing) |
| ECSAO | Study Abroad (Engineering & Computing) |
| FM | BSc in Financial & Actuarial Mathematics |
| HMSA | Study Abroad (Humanities & Soc Science) |
| HMSAO | Study Abroad (Humanities & Soc Science) |
| MS | BSc in Mathematical Sciences |
| SHSA | Study Abroad (Science & Health) |
| SHSAO | Study Abroad (Science & Health) |
| SMPSC | Single Module Professional Science |
| Archives: | |
