MA319-7-AU-CO:
Stochastic Processes
2020/21
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 08 October 2020
Friday 18 December 2020
15
16 July 2020
Requisites for this module
(none)
(none)
(none)
(none)
(none)
DIP GN1309 Mathematics and Finance,
MSC GN1312 Mathematics and Finance,
MSC GN1324 Mathematics and Finance,
DIP G20109 Optimisation and Data Analytics,
MSC G20312 Optimisation and Data Analytics,
DIP G30009 Statistics,
MSC G30012 Statistics,
DIP N32309 Actuarial Science,
MSC N32312 Actuarial Science,
MPHDG20048 Operational Research,
PHD G20048 Operational Research,
MPHDG30048 Statistics,
PHD G30048 Statistics,
MPHDN32348 Actuarial Science,
PHD N32348 Actuarial Science
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).
The aims of this module are:
1. To analyse in detail stochastic processes such as random walks, Markov processes, Poisson processes and birth and death processes.
2. To examine in detail the most important time series models.
3. To introduce the principles of actuarial modelling.
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)
• Use computational methods in R to solve and model stochastic processes
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 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.
Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.
Additional sessions will run for PG students.
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 |
Coursework weighting |
| Coursework |
Test |
|
|
| Exam |
Main exam: 240 minutes during Summer (Main Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Joseph Bailey, email: jbailef@essex.ac.uk.
Dr Joseph Bailey, Dr Tolulope Fadina & Dr Yanchun Bao
Dr Joseph Bailey (jbailef@essex.ac.uk), Dr Tolulope Fadina (t.fadina@essex.ac.uk), Dr Yanchun Bao (ybaoa@essex.ac.uk)
Yes
No
No
Dr Dimitrina Dimitrova
Cass Business School, City, University of London
Senior Lecturer
Available via Moodle
Of 3026 hours, 6 (0.2%) hours available to students:
3020 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).
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