Random Structures, Applied Probability and Computation
LMS Research School on Probability at the University of Liverpool
Taught Courses
Computational methods for financial Lévy models - Søren Asmussen
Abstract:
Lévy processes appear naturally as approximating continuous-time models of many discrete structures: they are thus universal among large classes of micro-models. Their numerical aspects have been explored intensely in financial mathematics models taking into account, in particular, higher steepness at the mode and heavier tails of the log returns compared to the Brownian Black-Scholes model, and the presence of jumps. When extended with regime-switching, they form a very flexible, even dense, model class. Computational aspects of Lévy models are, however, much more difficult than models based on stochastic differential equations with Brownian noise. In the course, we will, in particular, consider fitting methods, martingales for phase-type jumps, Gram-Charlier expansions, as well as simulation.
Course Material:
The course will require only probability theory at the master level. Students without previous exposure to Lévy processes, should in principle, be able to follow the course, but are recommended to prepare by reading Sections 1-3 of Chapter I (of the course notes in preparation): Here. These sections will only be covered quite quickly in the lectures. Previous knowledge of financial mathematics is not necessary either. Its present state is preliminary, so there is likely to be quite a few typos, inconsistency in notation etc.
Chapters I, II, III: Here
Slides: Day 1, Day 2 Part 1, Day 2 Part 2, Day 4
Exercises: Here
Remarks on Lévy Process Simulation: Here
References:
Here.
Self-similar Markov trees - Jean Bertoin
Abstract:
I will present some parts of an ongoing project jointly with Nicolas Curien (Paris-Sud) and Armand Riera (Paris-Sorbonne) about a rather large family of random objects which we call "Self-Similar Markov Trees". These objects consist of a quadruplet (T, d, m, g) where T is a rooted compact continuous tree, d the distance on T, m a finite Borel measure on T, and finally g: T --> R_{+} a "decoration". One of our main motivations is that such SSMT's arise as the scaling limits of Galton-Watson branching processes with integer types, and as such, in particular in a variety of models in discrete geometries.
Tentative Plan:
Overview of the mini-course
Topology of decorated trees and construction
Representation of general branching processes with real types
Self-similar Markov branching processes
Galton-Watson branching processes with integer types
Elements for a scaling limit theorem
Course Material:
Course notes on the topic by a similar mini-course taught by Nicolas Curien in the X Escuela de Probabilidad y Procesos Estocásticos (México, 2022): Here. This course will not follow the notes, but they may be useful.
References:
B. Haas, G. Miermont. "Scaling limits of Markov branching trees with applications to Galton– Watson and random unordered trees." The Annals of Probability, 40 (6), 2012.
F. Rembart, M. Winkel. "Recursive construction of continuum random trees." The Annals of Probability, 46 (5), September 2018.
Pólya urns - Cécile Mailler
Abstract:
In the first part of the course, we will review classical results on Pólya urns, from Markov’s results on the standard urn (1906), to Janson’s results on "irreducible" urns (2004). In the second part, we will focus on more recent developments of the theory, and in particular the generalisation to infinitely-many colours.
Course Material:
A feature wrote by Cécile for the LMS: Here. The course will be an extension of this.
Lecture Notes: Here
Exercises: Here (run by Christopher Dean, University of Bath), Solutions: Here
References:
Athreya, K. B., and P. E. Ney. "Branching Processes" (Chapter V). Springer Berlin, 1972.
R. van der Hofstad. "Random Graphs and Complex Networks". Cambridge Series in Statistical and Probabilistic Mathematics, 2017.
S. Janson. "Functional limit theorems for multitype branching processes and generalized Pólya urns". Stoch. Proc. App. 110 (2), 2004.
N.L. Johnson, S. Kotz. "Urn models and their application; an approach to modern discrete probability theory". Wiley, 1977.