Registry
Module Specifications
Archived Version 2012 - 2013
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Description FIM3 version of Stochastic Modelling : This module provides a comprehensive introduction to Stochastic Processes and their applications. 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 . | |||||||||||||||||||||||||||||||||||||
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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. | |||||||||||||||||||||||||||||||||||||
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 |
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Indicative Content and
Learning Activities Stochastic ModellingReview of basic probabilistic concepts, the various types of stochastic processes, stationarity, Markov processes, the Chapman-Kolmogorov equation, stationary probability distributions. [CT4 - (ii)]Markov ChainsSolution 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 ProcessesThe 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)] | |||||||||||||||||||||||||||||||||||||
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Indicative Reading List
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Other Resources None | |||||||||||||||||||||||||||||||||||||
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