In order to pass the course (grade 3) the student should
have a general knowledge of the theory of stochastic processes, in particular Markov processes, and be prepared to use Markov processes in various areas of applications;
be familiar with Markov chains in discrete and continuous time with respect to state diagram, recurrence and transience, classification of states, periodicity, irreducibility, etc., and be able to calculate transition probabilities and intensities;
be able to give an account of existence and uniqueness for stationary and asymptotic distributions of Markov chains and, whenever applicable, compute such distributions as solutions of a balance equation;
be able to calculate absorption probabilities and expected absorption time for Markov chains using the principle of conditioning with respect to the first jump;
be able to choose a suitable Markov model in various cases and make suitable calculations, in particular modelling of birth-death processes;
have a knowledge of Markov processes with a continuous state space, in particular a preparatory knowledge of Brownian motion and diffusion, and some understanding of the connection between the theory of Markov processes and differential equations;
have a knowledge of some general Markov method, e.g. Markov Chain Monte Carlo.
The Markov property. Chapman-Kolmogorov's relation, classification of Markov processes, transition probability. Transition intensity, forward and backward equations. Stationary and asymptotic distribution. Convergence of Markov chains. Birth-death processes. Absorption probabilities, absorption time. Brownian motion and diffusion. Geometric Brownian motion. Generalised Markov models. Applications of Markov chains.