To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for 'dynamic' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.

(LO1) Understand the theory of continuous-time Markov chains.

(LO2) Understand the theory of diffusion processes. 

(LO3) Be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes.

(LO4) Acquire an understanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

(S2) Numeracy

·Introduction and preliminaries: Conditional Probability/expectation with a continuous random variable, Conditional densities, Random processes; Continuous-time stochastic processes.
·Basic concepts of continuous-time Markov chains:  Definitions, Transition probabilities and their properties, Chapman-Kolmogorov equations, Absolute distributions; Transition rate matrix, Kolmogorov forward and backward equations.
·Properties of continuous-time Markov chains:  Classification of states, Absorbing state and probabilities, Recurrence and transience, Ergodicity, Limiting and stationary probabilities.
·Application and examples of continuous-time Markov chains: Finite-state chains, A selection of examples including some of the following: Pure birth process, Pure death process, Birth-and-death processes; M/M/1 queue and other Markovian queuing models, Simple branching processes; Application to epidemiology, Predator-prey processes, Competition proc esses; Population processes.
·Brownian motion processes: Multi-normal distributions, Standard BM, BMs with drift and their properties, Absolute distribution, Transition densities, Kolmogorov forward and backward equations, Co variance functions.