Stochastic Analysis Seminar
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Mon, 10/10/2005 14:15 |
Mr Max Skipper (Mathematical Institute, Oxford) |
Stochastic Analysis Seminar  |
DH 3rd floor SR |
| Numerous physical systems are justifiably modelled as Markov processes. However, in practical applications the (usually implicit) assumptions concerning accurate measurement of the system are often a fair departure from what is possible in reality. In general, this lack of exact information is liable to render the | |||
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Mon, 10/10/2005 15:45 |
Dr Pierre Tarres (Mathematical Institute, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| A self-interacting random walk is a random process evolving in an environment depending on its past behaviour. The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions. Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system. The purpose of our talk is to present our recent results on the subject [1,3,4,6]. Bibliography [1] M. Bena | |||
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Mon, 17/10/2005 14:15 |
Dr Peter M (University of Bath) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| /notices/events/abstracts/stochastic-analysis/mt05/m | |||
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Mon, 17/10/2005 15:45 |
Dr Alex Scott (Mathematical Institute, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Given a family of independent events in a probability space, the probability that none of the events occurs is of course the product of the probabilities that the individual events do not occur. If there is some dependence between the events, however, then bounding the probability that none occurs is a much less trivial matter. The Lov | |||
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Mon, 24/10/2005 14:15 |
Prof Alexander Grigoryan |
Stochastic Analysis Seminar |
Imperial College |
| I will present two-sided estimates for the heat kernel of the elliptic Schr | |||
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Mon, 24/10/2005 15:45 |
Professor Thomas Duquesne (Université de Paris-Sud, Orsay) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Levy trees are random continuous trees that are obtained as scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in PTRF) math.PR/0509518 (preprint) math.PR/0509690 (preprint). | |||
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Mon, 31/10/2005 14:15 |
Dr Chengui Yuan (University of Wales, Swansea) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence. | |||
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Mon, 31/10/2005 15:45 |
Dr Yuzuru Inahama (University of Osaka, Japan) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
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Mon, 07/11/2005 14:15 |
Prof Michel Ledoux (Université de Toulouse) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We examine the classical orthogonal polynomial ensembles using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. Equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Applications to sharp deviation inequalities on largest eigenvalues are discussed. | |||
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Mon, 07/11/2005 15:45 |
Dr Alexei Piunovskiy (The University of Liverpool) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| First of all, I intend to remind us of several properties of polyhedral cones and cone-generated orders which will be used for constructing Pareto sets in multiple objective optimisation problems. Afterwards, I will consider multiple objective discounted Markov Decision Process. Methods of Convex Analysis and the Dynamic Programming Approach allow one to construct the Pareto sets and study their properties. For instance, I will show that in the unichain case, Pareto sets for different initial distributions are topologically equivalent. Finally, I will present an example on the optimal management of a deteriorating system. | |||
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Mon, 14/11/2005 14:15 |
Mr Christian Litterer (Mathematical Institute, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
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Mon, 14/11/2005 15:45 |
Dr Huaizhong Zhao (University of Loughborough) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| In this talk I will present recent development of the generalized Ito's formulae of continuous semimartingales for less smooth function | |||
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Mon, 21/11/2005 14:15 |
Prof Shizan Fang (Université de Bourgogne) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| A classical result due to Kunita says that if the coefficients are global Lipschitzian, then the s.d.e defines a global flow of homeomorphisms. In this talk, we shall prove that under suitable growth on Lipschitz constants, the sde define still a global flow. | |||
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Mon, 21/11/2005 15:45 |
Dr M Tretyakov (University of Leicester) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients possessing unique solutions make up a very important class in applications. For instance, Langevin-type equations and gradient systems with noise belong to this class. At the same time, most numerical methods for SDEs are derived under the global Lipschitz condition. If this condition is violated, the behaviour of many standard numerical methods in the whole space can lead to incorrect conclusions. This situation is very alarming since we are forced to refuse many effective methods and/or to resort to some comparatively complicated numerical procedures. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of SDEs with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be made arbitrarily small with increasing the radius of the sphere. The results obtained are supported by numerical experiments. The concept of rejecting exploding trajectories is applied to computing averages with respect to the invariant law for Langevin-type equations. This approach to computing ergodic limits does not require from numerical methods to be ergodic and even convergent in the nonglobal Lipschitz case. The talk is based on joint papers with G.N. Milstein. | |||
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Mon, 28/11/2005 14:15 |
Dr Julien Berestycki (Marseille University) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999). These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the infinite site frequency spectrum associated with mutations in the context of population genetics. | |||
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Mon, 28/11/2005 15:45 |
Professor Milne Anderson (UCL) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Using the dyadic parametrization of curves, and elementary theorems and probability theory, examples are constructed of domains having bad properties on boundary sets of large Hausdorff dimension (joint work with F.D. Lesley). | |||
