The University of Southampton
Courses

# MATH6128 Stochastic Processes

## Module Overview

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.

### Aims and Objectives

#### Module Aims

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

### 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

Kulkarni VG (1999). Modelling, Analysis, Design and Control of Stochastic Systems.

Hickman JC (1997). Introduction to Actuarial Modelling. North American Actuarial Journal. ,1 , pp. pg. 1-5.

Karlin S & Taylor A (1975). A First Module in Stochastic Process.

Grimmett G (1992). Probability and Random Processes : Problems and Solutions.

Grimmett G & Stirzaker D (2001). Probability and Random Processes.

Brzezniak Z & Zastawniak T (1998). Basic Stochastic Processes : A Module Through Exercises.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

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