The module introduces stochastic processes, principles of actuarial modelling and time series models and analysis.
On completion of the course students should be able to (learning outcomes):
- Understand concepts of stochastic processes;
- Understand properties of Markov chain models for discrete-state processes;
- Understand applications of Poisson processes;
- Understand basic concepts to model and to analyse time series
- Understand queue models and their properties.
General stochastic process models. Random walks. Reflecting and absorbing barriers. Mean recurrence time, mean time to absorption. Difference equations. Branching processes. Markov chain models for discrete-state processes. Transition matrices: 1-step and n-step. Classification of states. Equilibrium distributions for time-homogeneous chains. Detail balance, general balance, limiting distribution, stationary distribution. Poisson processes. Differential-difference equations.
Birth and death processes.
Queues. The M/M/1 queue. Differential-difference equations. Conditions for equilibrium. Equilibrium distributions of queue size and waiting time for first-come-first-served queues. Extensions to M/M/k and M/M/ queues. The M/G/1 queue, imbedded Markov chain analysis. The Pollaczek-Khintchine formula. Mean queue length and waiting time.
Time series models; trend and seasonality. Stationarity. Autocovariance, autocorrelation and partial autocorrelation functions. Correlograms. Autoregressive (AR) processes. Moving average (MA) processes. ARMA processes. ARIMA processes and Box-Jenkins methods. Forecasting and minimising expected prediction
variance. Introduction to frequency domain analysis. Spectral density function. Periodograms.