Probability theory and Real Analysis
Students should:
1. Develop the skills to model simple real problems and propose a solution;
2. Solve theoretical problems, using the appropriate mathematical tools;
3. Read the related texts and gain access to more advanced courses;
4. Get a first flavour of the relevant research problems.
1. Discrete time processes: Markov chains in finite and countable space, limiting distribution;
2. Continuous time processes: density and distribution of into-event time for Poisson process, applications and extensions: e.g. birth-and-death processes, queues, epidemics;
3. Renewal processes: ordinary renewal process, renewal theorem, equilibrium
renewal process, application to queues;
4. Wiener processes and basic stochastic calculus: basic definitions and properties, It\^o's formula, Stochastic Differential Equations.
1. Markov Chains, J.R. Norris, Cambridge University Press;
2. Introduction to Stochastic Processes, G. Lawler, Chapman & Hall;
3. Basic Stochastic Processes, A Course Through Exercises, Z. Brzezniak and T. Zastawniak, Springer;
4. Probability and Random Processes, G. Grimmett and D. Stirzaker, 3rd Edition, Oxford University Press;
5. A first look at Rigorous Probability Theory, J. Rosenthal, World Scientific.