## MA319-6-AU-CO:Stochastic Processes

The details
2024/25
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Current
Thursday 03 October 2024
Friday 13 December 2024
15
16 May 2024

Requisites for this module
MA108 and MA200
(none)
(none)
(none)

MA322

## Key module for

BSC N233 Actuarial Science (Including Placement Year),
BSC N233DT Actuarial Science (Including Placement Year),
BSC N323 Actuarial Science,
BSC N323DF Actuarial Science,
BSC N323DT Actuarial Science,
BSC N324 Actuarial Science (Including Year Abroad),
BSC N325 Actuarial Science (Including Foundation Year),
BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year),
BSC I1G3 Data Science and Analytics,
BSC I1GB Data Science and Analytics (Including Placement Year),
BSC I1GC Data Science and Analytics (Including Year Abroad),
BSC I1GF Data Science and Analytics (Including Foundation Year),
MSCIN399 Actuarial Science and Data Science,
BSC N333 Actuarial Studies,
BSC N333DT Actuarial Studies,
BSC N334 Actuarial Studies (Including Placement Year),
BSC N334DT Actuarial Studies (Including Placement Year),
BSC N335 Actuarial Studies (Including Year Abroad)

## Module description

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

## Module aims

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.

## Module learning outcomes

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

1. Describe and classify stochastic processes (CS2 SO - 3.1)

2. Understand concepts underlying time series models (CS2 SO- 1)

3. Apply time series models (CS2 SO - 2.2)

4. Define and apply a Markov chain (CS2 SO - 3.2)

5. Define and apply a Markov process (CS2 SO - 3.3)

6. Derive maximum likelihood estimators for transition intensities (CS2 SO - 4.3)

7. Use computational methods in R to analyse and model stochastic processes (CS2B)

## Module information

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

## Learning and teaching methods

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.

## Bibliography*

The above list is indicative of the essential reading for the course.
The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

## Assessment items, weightings and deadlines

Coursework / exam Description Deadline Coursework weighting
Coursework   Test
Exam  Main exam: In-Person, Open Book (Restricted), 180 minutes during Summer (Main Period)
Exam  Reassessment Main exam: In-Person, Open Book (Restricted), 180 minutes during September (Reassessment 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.

Coursework Exam
30% 70%

### Reassessment

Coursework Exam
30% 70%
Module supervisor and teaching staff
Dr Igor Rodionov, email: igor.rodionov@essex.ac.uk.
Dr Igor Rodionov
maths@essex.ac.uk

Availability
Yes
Yes
No

## External examiner

Dr Murray Pollock
Newcastle University
Director of Statistics / Senior Lecturer
Resources
Available via Moodle
Of 39 hours, 35 (89.7%) hours available to students:
4 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.

Further information

* Please note: due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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