Each day there were hours of lectures plus exercise sessions. The lectures by J. Le Gall dealt more specifically with the coding of the genealogy of continuous branching processes, including applications to limit theorems for discrete Galton-Watson trees and to the construction of superprocesses. Part A J. Bertoin Preliminaries on Poisson measure, compensation and exponential formulas.
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Each day there were hours of lectures plus exercise sessions. The lectures by J. Le Gall dealt more specifically with the coding of the genealogy of continuous branching processes, including applications to limit theorems for discrete Galton-Watson trees and to the construction of superprocesses.
Part A J. Bertoin Preliminaries on Poisson measure, compensation and exponential formulas. Examples, connections with excursions of Markov processes, regenerative sets and local times. Passage times, renewal theory and Dynkin-Lamperti theorem. Fluctuation theory: Markov property of the reflected process, duality lemma and calculation of the exponents. The scale function and its applications two-sided exit problem, points of increase Continuous state branching processes.
Part B J. Le Gall Discrete and continuous branching processes. The construction of superprocesses and their Laplace functionals. The quadratic branching case. The Brownian snake approach to quadratic superprocesses.
The connection with branching processes: A generalized Ray-Knight theorem. Limit theorems for Galton-Watson trees and applications to reduced trees. The genealogical structure of general continuous trees. Applications to superprocesses. Cambridge Tracts in Mathematics, Cambridge University Press, Lectures in Mathematics. This mini-course will, in particular, treat applications to financial economics.
Furthermore there were the following two survey lectures: Sergei Levendorskii Rostov-on-Don : Option pricing under Levy processes and boundary problems for pseudo-differential operators. Schedule Note: really means So some of the talks may be related to the subject of the course, whereas others may focus on other areas of stochastics.
Schedule Monday 4 September ; in Auditorium G2 Recent empirical studies show that the normal inverse Gaussian distribution is a very good and flexible model for logreturns capturing non-Gaussian effects like, e. This distribution leads to a geometric pure-jump Levy process dynamics of the stock prices.
We treat a portfolio optimization problem where the investor derives her utility from present and past consumption through Hindy-Huang-Kreps preferences. The value function of the singular stochastic control problem is shown to be the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation, which will be an integro-differential equation subject to gradient constraints in our case. We study markets both with and without transaction costs, and derive explicit solutions in some special cases.
ABSTRACT: In order to capture key features of stock returns Barndorff-Nielsen and Shephard have introduced a new class of stochastic volatility models characterized by the use of processes of the Ornstein-Uhlenbeck type and allowing for a leverage effect. In this work we discuss these models from the viewpoint of derivative asset analysis.
For each of these possible pricing measures Q, a closed form formula for the price p Q of a European call option is determined. Finally, for several concrete examples we discuss different numerical approaches to the actual computation of option values. No assumptions of boundedness or boundedness away from zero are imposed We introduce a class of points which will be called isolated singular points, and investigate the weak existence as well as the uniqueness of the solution in the neighborhood of such a point.
A complete qualitative classification if these points is presented: there are 63 different types. It has been found that, for 59 types, there exists a unique solution in the neighborhood of an isolated singular point. This solution is defined up to the moment it leaves some interval.
Moreover, the solution is a strong Markov process. The remaining 4 types of isolated singular points we call them branch types disturb the uniqueness. In particular, there exist non-Markov solutions. Voiculescu U. This concept may be encountered as the asymptotic relation between independent Gaussian random matrices, as the size of the matrices increase to infinity, and thus free probability provides a concrete model for the joint asymptotic behavior of independent random matrices.
The first part of the talk will be a short introduction to free probability, and subsequently I shall focus on the free version of the theory of self-decomposability. The talk is on joint work with O. Tuesday 5 September ; in Auditorium G2 before lunch; in D2 after lunch 9. Its fundamental solution is the probability density evolving in time governing the modelled stochastic process.
Convergent and asymptotic series for its approximation are given. ABSTRACT: Four types of random walk models of Markov type in one space-dimension are considered and their interrelations via passages to the limit of vanishing space or time step separately or in a correctly scaled manner simultaneously in space and time are considered.
The transition probabilities are chosen in the domain of attraction of Levy stable probability distributions so that these random walks approximate Levy-Feller diffusion processes that are governed by a pseudo-differential evolution equation generalizing the classical diffusion equation. Finally, a sketch is presented how to generalize the theory to random walks with memory, thus approximating diffusion processes that are fractional also in time.
The following notes were used for the course: Jean Bertoin:.
JEAN BERTOIN LEVY PROCESSES PDF
Local time and excursions of a Markov process; 5. Local times of a Levy process; 6. Elements of potential theory; 3. In sum, this will become the standard reference on the subject for all working probability theorists. Dispatched from the UK in 3 business days When will my order arrive?
ISBN 13: 9780521646321
BERTOIN LEVY PROCESSES PDF