Course co-ordinator(s): Prof Sergey Foss (Edinburgh), Dr Alistair Wallis (Edinburgh), Chew Chun Yong (Malaysia).
Aims:
To introduce fundamental stochastic processes which are useful in insurance, investment and stochastic modelling, and to develop techniques and methods for analysing the long term behaviour of these processes.
Summary:
In this course, we develop methods for modelling systems or quantities which change randomly with time. Specifically, the evolution of the system is described by a collection {Xt} of random variables, where Xt denotes the state of the system at time t.
Discrete-time processes studied include (renewal processes and) Markov chains. In particular we consider branching processes, random walk processes, and more general countable state-space chains.
Continuous-time processes studied include Poisson and compound Poisson processes; continuous time Markov processes; population, queueing and risk models.
Detailed Information
Course Description: Link to Official Course Descriptor.
Pre-requisite course(s): F78PA Probability and Statistics A & F78PB Probability and Statistics B .
Location: Edinburgh, Malaysia.
Semester: 1.
Syllabus:
- Review of independence
- Sequences of random variables and the Markov property
- Review of matrix algebra
- Review of summation notation and other useful concepts.
- Using the Markov property
- Absorbing Markov chains with finite state space:
- Computing probability of absorption
- Computing expected time to absorption
- First-step (backwards) equations
- Basic examples: Mouse and cheese, drunkard’s walk, Ehrenfest chain, genetic models, gambling chains, etc.
- Stationarity problem for finite space chains
- Tricks on the computation of the stationary distribution (fluxes, reversibility, symmetry)
- State classification
- Periodicity
- Convergence to stationarity
- The strong theorem (law) of large numbers for sequences of i.i.d. random variables
- Frequencies
- Simple random walk (SRW) in the integers
- Markov chains with infinite but countable state space
- Simulation of Markov Chains and Markov Processes
- Recurrence and transience
- Examples: Discrete option pricing, branching processes, insurance loss, queueing for service, waiting for a bus, etc.–as time permits
- Basic relations between exponential, gamma, Poisson and uniform distributions
- Simple point processes, Poisson and compound Poisson processes
- Continuous time Markov processes
- Numerical solution of the Kolmogorov Forward Equations for a time-inhomogeneous Markov process
- Examples
Reading list:
Useful reference books are:
- Grinstead and Snell. Introduction to Probability. American Mathematical Society
- P. Bremaud (1997). An Introduction to Probabilistic Modeling. Springer.
- P. Bremaud (1999). Markov Chains. Springer.
- J. R. Norris (1998). Markov Chains. Cambridge University Press.
- G. R. Grimmett & D. R. Stirzaker (2001) Probability and Random Processes, 3rd ed. Oxford University Press.
Recommended textbooks on background matters:
- D. Stirzaker (1999). Probability and Random Variables: a beginner’s guide, Cambridge University Press.
- K.L. Chung and F. Aitsahlia (2003). Elementary Probability Theory, Springer-Verlag.
- G. Grimmett & D. Welsh (1990). Probability: an Introduction, Oxford University Press.
- S. M. Ross (1988). A First Course in Probability, 3rd edition, Macmillan.
- D. Williams (2001). Weighing the Odds: A Course in Probability and Statistics, Cambridge University Press.
SCQF Level: 9.
Other Information
Help: If you have any problems or questions regarding the course, you are encouraged to contact the course leader.
Canvas: further information and course materials are available on Canvas