Forthcoming events in this series


Mon, 15 Jan 2007
15:45
DH 3rd floor SR

The Global Error in Weak Approximations of Stochastic Differential Equations

Dr Saadia Ghazali
(Imperial College London)
Abstract

In this talk, the convergence analysis of a class of weak approximations of

solutions of stochastic differential equations is presented. This class includes

recent approximations such as Kusuoka's moment similar families method and the

Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate

of convergence depends intrinsically on the smoothness of the chosen test

function. For smooth functions (the required degree of smoothness depends on the

order of the approximation), an equidistant partition of the time interval on

which the approximation is sought is optimal. For functions that are less smooth

(for example Lipschitz functions), the rate of convergence decays and the

optimal partition is no longer equidistant. An asymptotic rate of convergence

will also be presented for the Lyons-Victoir method. The analysis rests upon

Kusuoka-Stroock's results on the smoothness of the distribution of the solution

of a stochastic differential equation. Finally, the results will be applied to

the numerical solution of the filtering problem.

 

Mon, 15 Jan 2007
14:15
DH 3rd floor SR

Differential Equations Driven by Gaussian Signals

Dr Peter Fritz
(University of Cambridge)
Abstract

We consider multi-dimensional Gaussian processes and give a novel, simple and

sharp condition on its covariance (finiteness of its two dimensional rho-variation,

for some rho <2) for the existence of "natural" Levy areas and higher iterated

integrals, and subsequently the existence of Gaussian rough paths. We prove a

variety of (weak and strong) approximation results, large deviations, and

support description.

Rough path theory then gives a theory of differential equations driven by

Gaussian signals with a variety of novel continuity properties, large deviation

estimates and support descriptions generalizing classical results of

Freidlin-Wentzell and Stroock-Varadhan respectively.

(Joint work with Nicolas Victoir.)

 

Mon, 20 Nov 2006
14:15
DH 3rd floor SR

Branching Markov Chains

Professor Nina Gantert
(Universitat Munster)
Abstract

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Mon, 13 Nov 2006
15:45
DH 3rd floor SR

Randon tilings and random matrices

Professor Kurt Johansson
(KTH Stockholm)
Abstract

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Mon, 06 Nov 2006
15:45
L1

Pathwise stochastic optimal control

Professor Chris Rogers
(University of Cambridge)
Abstract
 

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Mon, 30 Oct 2006
14:15
DH 3rd floor SR

The ensemble Kalman filter: a state estimation method for hazardous weather prediction

Dr Sarah Dance
(University of Reading)
Abstract
Numerical weather prediction models require an estimate of the current state of the atmosphere as an initial condition. Observations only provide partial information, so they are usually combined with prior information, in a process called data assimilation. The dynamics of hazardous weather such as storms is very nonlinear, with only a short predictability timescale, thus it is important to use a nonlinear, probabilistic filtering method to provide the initial conditions. 

Unfortunately, the state space is very large (about 107 variables) so approximations have to be made.

 The Ensemble Kalman filter (EnKF) is a quasi-linear filter that has recently been proposed in the meteorological and oceanographic literature to solve this problem. The filter uses a forecast ensemble (a Monte Carlo sample) to estimate the prior statistics. In this talk we will describe the EnKF framework and some of its strengths and weaknesses. In particular we will demonstrate a new result that not all filters of this type bear the desired relationship to the forecast ensemble: there can be a systematic bias in the analysis ensemble mean and consequently an accompanying shortfall in the spread of the analysis ensemble as expressed by the ensemble covariance matrix. This points to the need for a restricted version of the notion of an EnKF. We have established a set of necessary and sufficient conditions for the scheme to be unbiased. Whilst these conditions are not a cure-all and cannot deal with independent sources of bias such as modelling errors, they should be useful to designers of EnKFs in the future.

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Mon, 23 Oct 2006
14:15
DH 3rd floor SR

Dual Nonlinear Filters and Entropy Production

Dr Nigel Newton
(University of Essex)
Abstract
The talk will describe recent collaborative work between the speaker and Professor Sanjoy Mitter of MIT on connections between continuous-time nonlinear filtering theory, and nonequilibrium statistical mechanics. The study of nonlinear filters from a (Shannon) information- theoretic viewpoint reveals two flows of information, dubbed 'supply' and 'dissipation'. These characterise, in a dynamic way, the dependencies between the past, present and future of the signal and observation processes. In addition, signal and nonlinear filter processes exhibit a number of symmetries, (in particular they are jointly and marginally Markov), and these allow the construction of dual filtering problems by time reversal. The information supply and dissipation processes of a dual problem have rates equal to those of the original, but with supply and dissipation exchanging roles. The joint (signal-filter) process of a nonlinear filtering problem is unusual among Markov processes in that it exhibits one-way flows of information between components. The concept of entropy flow in the stationary distribution of a Markov process is at the heart of a modern theory of nonequilibrium statistical mechanics, based on stochastic dynamics. In this, a rate of entropy flow is defined by means of time averages of stationary ergodic processes. Such a definition is inadequate in the dynamic theory of nonlinear filtering. Instead a rate of entropy production can be defined, which is based on only the (current) local characteristics of the Markov process. This can be thought of as an 'entropic derivative'. The rate of entropy production of the joint process of a nonlinear filtering problem contains an 'interactive' component equal to the sum of the information supply and dissipation rates. These connections between nonlinear filtering and statistical mechanics allow a certain degree of cross- fertilisation between the fields. For example, the nonlinear filter, viewed as a statistical mechanical system, is a type of perpetual motion machine, and provides a precise quantitative example of Landauer's Principle. On the other hand, the theory of dissipative statistical mechanical systems can be brought to bear on the study of sub-optimal filters. On a more philosophical level, we might ask what a nonlinear filter can tell us about the direction of thermodynamic time.    
Mon, 16 Oct 2006
15:45
DH 3rd floor SR

5x+1: how many go down?

Dr Stanislav Volkov
(University of Bristol)
Abstract

 

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Mon, 16 Oct 2006
14:15
DH 3rd floor SR

TBA

Prof Liming Wu
(Universite Blaise Pascal-Clermont-Ferrand II)
Mon, 29 May 2006
15:45
DH 3rd floor SR

TBA

Michael Caruana
(Oxford)
Mon, 22 May 2006
14:15
DH 3rd floor SR

Exotic couplings of Brownian motion

Prof Wilfrid Kendall
(University of Warwick)
Abstract

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Mon, 08 May 2006
15:45
DH 3rd floor SR

TBA

Prof Feng-Yu Wang
(Beijing Normal University)
Abstract

TBA

Mon, 06 Mar 2006
17:00
DH 3rd floor SR

to be arranged

Prof Dominique Bakry
(Université de Toulouse)
Mon, 27 Feb 2006
15:45
DH 3rd floor SR

Title to be confirmed

Dr Qingyang Guan
(Dept of Statistics, Oxford)
Mon, 23 Jan 2006
14:15
DH 3rd floor SR

Limit theorems for subsequences of random variables

Professor Sergey Bobkov
(University of Minnesota)
Abstract
We will be discussing limit behaviour of sums along subsequences of a given sequence of non-correlated random variables. Some results are applied to the classical trigonometric system in the Berkes model. /notices/events/abstracts/stochastic-analysis/ht06/bobkov.shtml