The University of Southampton
Courses

# MATH2012 Stochastic Processes

## Module Overview

The module will introduce the basic ideas in modelling, solving and simulating stochastic processes.

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Understand the definition of a stochastic process and in particular a Markov process, a counting process and a random walk
• Recall the definition and derive some basic properties of a Poisson process
• State the Kolmogorov equations for a Markov process where the transition intensities depend not only on age/time, but also on the duration of stay in one or more states
• Demonstrate how a Markov jump process can be simulated
• Derive the Kolmogorov equations for a Markov process with time independent and time/age dependent transition intensities
• Understand survival, sickness and marriage models in terms of Markov processes
• Understand, in general terms, the principles of stochastic modelling
• Define and explain the basic properties of Brownian motion, demonstrate an understanding of stochastic differential equations and then to integrate, and apply the Ito formula
• Calculate the distribution of a Markov chain at a given time
• Demonstrate how a Markov chain can be simulated
• Classify a stochastic process according to whether it operates in continuous or discrete time and whether it has a continuous or a discrete state space, and give examples of each type of process
• Classify the states of a Markov chain as transient, null, recurrent, positive recurrent, periodic, aperiodic and Ergodic
• Describe a time-inhomogeneous Markov chain and its simple applications
• Describe a Markov chain and its transition matrix
• Determine the stationary and equilibrium distributions of a Markov chain
• Solve the Kolmogorov equations in simple cases

### Syllabus

Markov Chain Definition and basic properties Classification of states and decomposition of state space The long term probability distribution of a Markov chain Modelling using Markov chains Time-homogeneous Markov jump process Poisson process and its basic properties Birth and death processes Kolmogorov differential equations Structure of a Markov jump process Time-inhomogeneous Markov jump process Definition and basics A survival model A sickness and death model A marriage model Sickness and death with duration dependence Basic principles of stochastic modelling Classification of stochastic modelling Postulating, estimating and validating a model Simulation of a stochastic model and its applications Brownian motion: Definition and basic properties. Stochastic differential equations, the Ito integral and Ito formula. Diffusion and mean testing processes. Solution of the stochastic differential equation for the geometric Brownian motion and Ohrnstein-Uhlenbeck process

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem classes, coursework, surgeries and private study

TypeHours
Independent Study102
Teaching48
Total study time150

GRIMMETT G and STIRZAKER D (2001). Probability and random processes.

GRIMMETT G (1992). Probability and random processes: problems and solutions.

BRZEZNIAK Z and ZASTAWNIAK T (1998). Basic Stochastic Processes: a course through exercises.

KULKARNI V G (1999). Modelling, analysis, design and control of stochastic systems.

HICKMAN J C (1997). Introduction to actuarial modelling. North American Actuarial Journal. ,1 , pp. pg.1-5.

KARLIN S and TAYLOR A (1975). A first course in stochastic process.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 30%
Written exam 70%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

Pre-requisites: MATH2011 OR ECON2006

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.