The module will introduce the basic ideas in modelling, solving and simulating stochastic processes.
Aims and Objectives
Having successfully completed this module you will be able to:
- Describe a Markov chain and its transition matrix
- 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
- Demonstrate how a Markov jump process can be simulated
- Understand the definition of a stochastic process and in particular a Markov process, a counting process and a random walk
- 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
- 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
- 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 chain can be simulated
- Understand survival, sickness and marriage models in terms of Markov processes
- Understand, in general terms, the principles of stochastic modelling
- Determine the stationary and equilibrium distributions of a Markov chain
- Recall the definition and derive some basic properties of a Poisson process
- Solve the Kolmogorov equations in simple cases
- Derive the Kolmogorov equations for a Markov process with time independent and time/age dependent transition intensities
- Calculate the distribution of a Markov chain at a given time
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
|Total study time||150|
Resources & Reading list
Hickman JC (1997). Introduction to Actuarial Modelling. North American Actuarial Journal, 1(3), pp. pg. 1-5.
Kulkarni VG (1999). Modelling, Analysis, Design and Control of Stochastic Systems. Springer.
Karlin S & Taylor A (1975). A First Module in Stochastic Process. Academic Press.
Brzezniak Z & Zastawniak T (1998). Basic Stochastic Processes : A Module Through Exercises. Springer.
Grimmett G (1992). Probability and Random Processes : Problems and Solutions. Oxford University Press.
Grimmett G & Stirzaker D (2001). Probability and Random Processes. Oxford University Press.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External