# Past Stochastic Analysis Seminar

1 November 2004
14:15
Professor Y M Suhov
Abstract
Anderson localisation is an important phenomenon describing a transition between insulation and conductivity. The problem is to analyse the spectrum of a Schroedinger operator with a random potential in the Euclidean space or on a lattice. We say that the system exhibits (exponential) localisation if with probability one the spectrum is pure point and the corresponding eigen-functions decay exponentially fast. So far in the literature one considered a single-particle model where the potential at different sites is IID or has a controlled decay of correlations. The present talk aims at $N$-particle systems (bosons or fermions) where the potential sums over different sites, and the traditional approach needs serious modifications. The main result is that if the `randomness' is strong enough, the $N$-particle system exhibits localisation. The proof exploits the muli-scale analysis scheme going back to Froehlich, Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No preliminary knowledge of the related material will be assumed from the audience, apart from basic facts. This is a joint work with V Chulaevsky (University of Reims, France)
• Stochastic Analysis Seminar
25 October 2004
15:45
Professor Zhongmin Qian
Abstract
I will present a new formula for diffusion processes which involving Ito integral for the transition probability functions. The nature of the formula I discovered is very close to the Kac formula, but its form is similar to the Cameron-Martin formula. In some sense it is the Cameron-Martin formula for pinned diffusions.
• Stochastic Analysis Seminar
25 October 2004
14:15
Dr J Warren
Abstract
I will consider a stochastic process ( \xi_u; u \in \Gamma_\infty ) where \Gamma_\infty is the set of vertices of an infinite binary tree which satisfies some recursion relation \xi_u= \phi(\xi_{u0},\xi_{u1}, \epsilon_u) \text { for each } u \in \Gamma_\infty. Here u0 and u1 denote the two immediate daughters of the vertex u. The random variables ( \epsilon_u; u\in \Gamma_\infty), which are to be thought of as innovations, are supposed independent and identically distributed. This type of structure is ubiquitous in models coming from applied proability. A recent paper of Aldous and Bandyopadhyay has drawn attention to the issue of endogeny: that is whether the process ( \xi_u; u \in \Gamma_\infty) is measurable with respect to the innovations process. I will explain how this question is related to the existence of certain dynamics and use this idea to develop a necessary and sufficient condition [ at least if S is finite!] for endogeny in terms of the coupling rate for a Markov chain on S^2 for which the diagonal is absorbing.
• Stochastic Analysis Seminar
18 October 2004
15:45
Abstract
We shall review recent progress in the understanding of isoperimetric inequalities for product probability measures (a very tight description of the concentration of measure phenomeonon). Several extensions of the classical result for the Gaussian measure were recently derived by functional analytic methods.
• Stochastic Analysis Seminar
18 October 2004
14:15
Dr J Trashorras
Abstract
The Hopfield model took his name and its popularity within the theory of formal neural networks. It was introduced in 1982 to describe and implement associative memories. In fact, the mathematical model was already defined, and studied in a simple form by Pastur and Figotin in an attempt to describe spin-glasses, which are magnetic materials with singular behaviour at low temperature. This model indeed shows a very complex structure if considered in a slightly different regime than the one they studied. In the present talk we will focus on the fluctuations of the free energy in the high-temperature phase. No prior knowledge of Statistical mechanics is required to follow the talk.
• Stochastic Analysis Seminar
11 October 2004
15:45
Professor N H Bingham
Abstract
Joe Doob, who died recently aged 94, was the last survivor of the founding fathers of probability. Doob was best known for his work on martingales, and for his classic book, Stochastic Processes (1953). The talk will combine an appreciation of Doob's work and legacy with reminiscences of Doob the man. (I was fortunate to be a colleague of Doob from 1975-6, and to get to know him well during that year.) Following Doob's passing, the mantle of greatest living probabilist descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick man.
• Stochastic Analysis Seminar
11 October 2004
14:15
Professor Sylvie Meleard
Abstract
We are interested in a microscopic stochastic description of a population of discrete individuals characterized by one adaptive trait. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. Once this point process is in place, the quest for tractable approximations can follow different mathematical paths, which differ in the normalization they assume (taking limit on population size , rescaling time) and in the nature of the corresponding approximation models: integro or integro-differential equations, superprocesses. In particular cases, we consider the long time behaviour for the stochastic or deterministic models.
• Stochastic Analysis Seminar
14 June 2004
14:15
Chris Potter
Abstract
Complete stochastic volatility models provide prices and hedges. There are a number of complete models which jointly model an underlying and one or more vanilla options written on it (for example see Lyons, Schonbucher, Babbar and Davis). However, any consistent model describing the volatility of options requires a complex dependence of the volatility of the option on its strike. To date we do not have a clear approach to selecting a model for the volatility of these options
• Stochastic Analysis Seminar
7 June 2004
15:45
Arnaud de La Pradelle
Abstract
A version of Lyons theory of rough path calculus which applies to a subclass of rough paths for which more geometric interpretations are valid will be presented. Application will be made to the Brownian and to the (fractional) support theorem.
• Stochastic Analysis Seminar