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The University of Southampton
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MATH6128 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

Resources & Reading list

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

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

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.

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

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

Assessment

Summative

MethodPercentage contribution
Coursework 30%
Exam 70%

Referral

MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External

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