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

#### Module Aims

To introduce the basic ideas in modelling, solving and simulating stochastic processes. Stochastic processes are systems which change in accordance with probabilistic laws. Examples of such processes are the behaviour of the size of a population, queues, random walks, breakdown of machines. These are all situations which evolve with time according to certain probabilities. However, some are concerned with discrete time, such as the position of a random walk after each step, others with continuous time, like the length of a queue at a supermarket checkout at any time instant. All these are modelled and solved in the module. The mathematical tools used are chiefly difference equations, differential equations, matrices and generating functions. These will be briefly revised at appropriate places in the module.

#### 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, acounting 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/agedependent 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 Study96
Teaching54
Total study time150

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

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

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

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

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

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

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Written exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External