Stochastic Processes

The details
Mathematical Sciences
Colchester Campus
Postgraduate: Level 7
Thursday 03 October 2019
Saturday 14 December 2019
01 October 2019


Requisites for this module



Key module for

DIP G10109 Mathematics,
MSC G10112 Mathematics,
MSC G10124 Mathematics,
DIP GN1309 Mathematics and Finance,
MSC GN1312 Mathematics and Finance,
MSC GN1324 Mathematics and Finance,
DIP G20109 Statistics and Operational Research,
MSC G20312 Statistics and Operational Research,
DIP G30009 Statistics,
MSC G30012 Statistics,
DIP N32309 Actuarial Science,
MSC N32312 Actuarial Science

Module description

This module introduces stochastic processes, time series models and analysis. This module covers 45% (CS2 Units 5-9 & 13) of required material for the Institute and Faculty of Actuaries CS2 syllabus (Risk Modelling and Survival Analysis, Core Principles).

Module aims


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 what is meant by the Markov property in the context of a stochastic process and in terms of filtrations

2. 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 how Markov chains can be used as a tool for modelling and how they can be simulated

3. Markov jump processes [CS2-3.3]:
• State the essential features of a Markov 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
• Describe simple survival models, sickness models and marriage models in terms of Markov processes and describe other simple applications
• 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 sickness and marriage models in terms of duration dependent Markov processes and describe other simple applications
• Demonstrate how Markov jump processes can be used as a tool for modelling and how they can be simulated

4. 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.

5. 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.

Module learning outcomes

On completion of this module, students should be able to:

• Describe and classify stochastic processes (3.1)
• Understand concepts underlying time series models (2.1)
• Apply time series models (2.2)
• Define and apply a Markov chain (3.2)
• Define and apply a Markov process (3.3)
• Derive maximum likelihood estimators for transition intensities (4.3)

Module information

No additional information available.

Learning and teaching methods

The module consists of 25 lectures, 5 classes. In the summer term 3 revision lectures are given.


This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Written Exam Test 1
Written Exam Test 2
Exam 180 minutes during Summer (Main Period) (Main)

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr Junlei Hu, email, Dr Joe Bailey (
Dr Junlei Hu (



External examiner

Dr Dimitrina Dimitrova
Cass Business School, City, University of London
Senior Lecturer
Available via Moodle
Of 42 hours, 36 (85.7%) hours available to students:
6 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

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