Module Directory

Stochastic process are seen everywhere from biological to financial systems. This module introduces stochastic processes, Markov processes, time series models and analysis.

The aims of this module are:

• To introduce the concept of a stochastic processes


• To analyse in detail stochastic processes such as random walks, Markov processes, Poisson processes.


• To introduce time series and examine in detail important time series models.


• To demonstrate how stochastic processes can be simulated and data analysed using computer programmes.

By the end of this module, students will be expected to be able to:

1. Describe and classify stochastic processes (3.1)


2. Understand concepts underlying time series models (2.1)


3. Apply time series models (2.2)


4. Define and apply a Markov chain (3.2)


5. Define and apply a Markov process (3.3)


6. Derive maximum likelihood estimators for transition intensities (4.3)


7. Use computational methods in R to solve and model stochastic processes.


8. Develop a more thorough understanding of the properties of simple 1D random walk models, such as results using the Ballot Theorem.


9. Describe Time-reversible Markov processes and determine if a given process is time-reversible.


10. Use R to determine best fitting ARMA models and forecast time-series data from these models.

This module covers Units 2, 3 & 4.3 of required material for the Institute and Faculty of Actuaries CS2 syllabus (Risk Modelling and Survival Analysis, Core Principles).

Indicative syllabus

1. Stochastic processes [CS2-3.1]:
Define in general terms a stochastic process and in particular a counting process
Classify a stochastic process according to whether it:
a. Operates in continuous or discrete time
b. Has a continuous or a discrete state space; or
c. Is a mixed type
And give examples of each type of process
Describe possible applications of mixed processes
Explain the Markov property.

2. Random Walks [CS2-2.2]
State the definition of a discrete time 1D random walk
Derive the expectation and variance
Understand what processes can be modelled by a random walk
Gambler's Ruin problem.

3. Markov chains [CS2-3.2]:
State the essential features of a Markov chain model
State the Chapman-Kolmogorov equations that represent a Markov chain
Calculate the stationary distribution for a Markov chain in simple cases
Describe a system of frequency-based experience rating in terms of a Markov chain and describe other simple applications
Describe a time-inhomogeneous Markov chain model and describe simple applications
Demonstrate using 'R' how Markov chains can be used as a tool for modelling and how they can be simulated

4. Markov jump processes [CS2-3.3]:
State the essential features of a Markov jump process model
Define a Poisson process, derive the distribution of the number of events in a given time interval, derive the distribution of inter-event times, and apply these results.
Derive the Kolmogorov equations for a Markov process with time independent and time/age dependent transition intensities
Solve the Kolmogorov equations in simple cases
State the Kolmogorov equations for a model where the transition intensities depend no only on age/time, but also on the duration of stay in one or more state.
Describe simple applications
Demonstrate how Markov jump processes can be used as a tool for modelling and how they can be simulated

5. Estimation in the Markov Model [CS2-4.3]:
i. Describe an observational plan in respect of a finite number of individuals observed during a finite period of time, and define the resulting statistics, including the waiting times.
ii. Derive the likelihood function for constant transition intensities in a Markov model of transfers between states given the statistics in (i).
iii. Derive maximum likelihood estimators for the transition intensities in (ii) and state their asymptotic joint distribution
iv. State the Poisson approximation to the estimator in (iii) in the case of a single decrement.

6. Time series models [CS2-2.1,2.2]:
Explain the concept and general properties of stationary, I(0), and integrated, I(1), univariate time series
Explain the concept of a stationary random series
Explain the concept of a filter applied to a stationary random series
Know the notation for backwards shift operator, backwards difference operator, and the concept of roots of the characteristic equation of time series.
Explain the concepts and basic properties of autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) time series.
Explain the concept and properties of discrete random walks and random walks with normally distributed increments, both with and without drift
Explain the basic concept of a multivariate autoregressive model
Explain the concept of cointegrated time series
Show that certain univariate time series models have the Markov property and describe how to rearrange a univariate time series model as a multivariate Markov model
Computational modelling of stochastic processes
Use 'R' to simulate AR, MA and ARIMA times series.
Use 'R' to analyse potential time series data

Teaching in the department will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.