After successful completion of the course, students are able to understand the basic concepts underlying the theory of discrete-time and continuous-time Markov chains and discrete-time martingales. They are also able to apply these concepts in a variety of applications.
Stochastic processes are mathematical objects aimed at modelling random phenomena evolving in time. We will deal with two of the simplest, and at the same time most important, types of stochastic processes: Markov chains and martingales. We will also see these models at work in a wide variety of applications.
MARKOV CHAINS
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Basic theory
- (Strong) Markov property
- Communicating classes
- Hitting times
- Recurrence and transience
- Invariant distributions
- Convergence to equilibrium
- Time reversibility
- Ergodic theorem
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Further theory in continuous time
- Generator matrices
- Jump chains and holding times
- Explosion
- Forward and backward equations
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Applications
- Random walks
- Birth-and-death processes
- Moran model in population genetics
- PageRank algorithm
- Markov chain Monte Carlo (MCMC) methods
- Poisson processes
- Queuing systems
- Inspection paradox
MARTINGALES
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Basic theory
- Filtrations and adapted processes
- Martingales, supermartingales and submartingales
- Martingale transform
- Stopped martingales
- Doob's optional stopping theorem
- Doob's decomposition
- Maximal inequalities
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Asymptotic theory
- Almost sure convergence: Doob's forward convergence theorem
- Convergence of martingales in L^2
- Uniform integrability and convergence in L^1
- Levy's upward theorem
- Doob's L^p inequality and convergence in L^p
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Applications
- Games and gambling strategies
- Kolmogorov's zero-one law
- Branching processes
- Statistical hypothesis testing
- Filtering problems in statistics
- Secretary problem
In principle, this VO course should run for 6 hours per week, from the beginning of March to the beginning of May. In any case, the schedule is provisional and will be discussed with the students in the first week (for example, we may change date/time or run it for fewer weekly hours over a longer period, e.g. until the end of May). If the provisional schedule does not work for you, please drop an email to the lecturer before the start of the semester and indicate your preference.
Written exam. Additional points may be awarded for class participation.
The first exam will take place after the end of the lectures, on a date to be agreed with attending students. If you wish to take the exam in a subsequent date, please get in touch directly with the lecturer and another exam will be scheduled.
