Summer term Program, 2009
Tuesday 28 April (4.10pm) (Please note different time)
Joint AMMP Colloquium – Stochastic Analysis seminar
Nicolas Dirr
Title of the talk: Interfaces in heterogeneous media
Abstract: I consider parabolic PDEs for the evolution of an interface with an additive periodic or random perturbation (modelling the interaction with a heterogeneous environment) and a constant forcing (driving field). I will present some results on the effective velocity of the interface on large scales for weak periodic forcing, and some results in the case of random forcing, with particular emphasis on the difference between the periodic and the random case.
Monday 22 June (Please note different day of the week and time)
Florin Avram
Title of the talk: Some Examples of Asymptotic Approximations for the Stationary Distribution of Queueing Networks
Abstract: We review the results of Ignatyuk, Malyshev, and Scherbakov (1994), and Mogulskii and Borovkov (2001) on the large deviations asymptotics of random walks on the orthant Z+I and present some examples where sharp asymptotics are also available.
Tuesday 30 June
Vassili Kolokoltsov
Title of the talk: SDEs driven by nonlinear Levy noise
Abstract: SDEs driven by nonlinear distribution dependent L’evy noise are introduced and studied. As an application, it is shown that a conditionally positive integro-differential operator (of the L’evy-Khintchine type) with variable coefficients (diffusion, drift and L'evy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metrized by the Wasserstein-Kantorovich metrics. This is a nontrivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.
Spring term Program, 2009
Tuesday 27 January
Title of the talk: Parameter Estimation for Multiscale Diffusions
Abstract: We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single-scale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly.
We first study this problem in the context of thermally activated motion in a two-scale potential. We then show how our results can be extended to cover the problem of fitting an averaged or homogenized equation to multiscale data, in maximum likelihood framework.
Tuesday 10 February
Andreas Kyprianou (Bath)
Title of the talk: Refracted Levy Processes
Abstract: We discuss solutions to a very elementary, but none the less degenerate, SDE which describes the aggregate path of a Levy process when is perturbed by a linear drift every time it spends time above a fixed level. Despite the simple nature of the SDE, some work is required to establish existence and uniqueness of a solution. This problem is put in context by an application in insurance mathematics.
Tuesday 17 February (4.10pm, Room 130) (Please note different time and location)
Joint AMMP Colloquium – Stochastic Analysis seminar
Peter Kloeden (Goethe Universität, Frankfurt)
Title of the talk: Random attractors and the preservation of synchronization in the presence of noise
Abstract: It is shown that the synchronization of dissipative systems involving one-sided dissipative Lipschitz conditions persists when they are disturbed by additive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable. For linear multiplicative noise the synchronization is modulo exponential factors involving Ornstein-Uhlenbeck processes corresponding to the driving noises. In all cases the SDE are transformed to corresponding random ordinary differential equations for which pathwise estimates can be obtained. The theory of random dynamical systems is used to established existence of the limiting solutions. Synchronization of stochastic reaction diffusion equations on thin domains separated by a permeable membrane will also be discussed.
Monday 23 February (Room 341) (Please note different day of the week and room)
Konstantinos Manolarakis (BNP Paribas)
Title of the talk: Solving a Backward SDE with the Cubature method
Abstract: By considering Backward Stochastic Differential Equations (BSDE) where the terminal condition is of the form ©(XT), where X is a diffusion, we are able to extend the well known Feynman-Kac formula to semi linear PDEs. Hence, probabilistic methods for the solution of BSDEs provide us with a new approach to the problem of approximating the solution of a semi-linear PDE.
Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons and Victoir. When the function © is at least Lipschitz continuous, we are able to recover satisfactory error estimates. We present numerical experiments that validate the method in both linear and non linear set ups.
Tuesday 3 March
Peter Fritz (Cambridge)
Title of the talk: A (rough) pathwise approach to a class of fully non-linear SPDEs
Abstract: We return to seminal work of P.L.Lions and P.Souganidis on nonlinear stochastic partial differential equations in viscosity sense and present some evidence that rough path analysis a la T.Lyons may allow to continue, and perhaps complete, the program they started in a series of papers from 1998-2003.
Tuesday 17 March
Mathew Penrose (Bath)
Title of the talk: Normal approximation in stochastic geometry
Abstract: Many quantities of interest in stochastic geometry can be expressed in terms of n random points in a window of size n. Under loca l dependence criteria, general central limit theorems for such quantities are known. In this talk we discuss recent work demonstrating refinements such as Berry-Esseen bounds and local CLTs. Examples include coverage processes and random geometric graphs.
Tuesday 24 March
Title of the talk: Ornstein-Uhlenbec processes with Levy noise
Abstract: The talk is concerned with finite and infinite dimensional Ornstein-Uhlenbeck processes perturbed by Levy noise. We discuss conditions under which the processes have densities and satisfy the strong Feller property. Regularizing properties of infinite dimensional processes are investigated as well. The results are based on joint research with E. Priola and Z. Brzezniak.
Thursday 2 April 2pm (Please note different day of the week and time)
Szymon Peszat (Institute of Mathematics of the Polish Academy of Sciences)
Title of the talk: Limit theorems for additive functional of alpha stable processes
Abstract: The t alk is concerned with a law of large numbers and functional central limit theorems for a class of additive functionals of alpha stable processes.
Autumn Term Program, 2008
Tuesday 14 October
James Norris (Cambridge)
Title of the talk: Planar aggregation and the coalescing Brownian flow
Abstract: A simple model for the random aggregation of particles in two dimensions can be formulated in terms of an iteration of random conformal maps. We show that, in the limit of small particle size and large particle numbers, the size of the fingers of the resulting cluster, as measured by their harmonic measure, evolves according to Arratia's coalescing Brownian flow. This is joint work with Amanda Turner.
Tuesday 21 October
Adam Ostaszewski (London School of Economics)
Title of the talk: Inference from Non-Disclosure
Abstract: Shin (2006) has argued that in order to understand the equilibrium patterns of corporate disclosure, it is necessary for researchers to work within an asset pricing model framework in which corporate disclosures are endogenously determined. Furthermore, he argues that without such a framework optimal disclosure strategies may seem counter-intuitive. With this in mind, we generalize the Dye (1985) and Penno (1997) upper tailed disclosure models, so that management's strategic disclosure behaviour can be shown to result in an optimal observable disclosure intensity. We show why a higher equilibrium disclosure intensity may need to be interpreted as implying management have less precise forecasts of future firm value (or, as referred to in the title, there is less precision in management's vision). The derived results call into question the specification of empirical studies which test whether firms with higher disclosure intensity will face a lower cost of capital. Working within a generalized Dye-Penno framework this research shows why in equilibrium the converse case applies.
Wednesday 22 October 2pm (Room 140) (Please note different day of the week, time and location)
Serge Cohen (Toulouse)
Title of the talk: Spectral measure of Brownian field on hyperbolic plane
Abstract: Brownian field on hyperbolic plane is a Gaussian field with stationary increments and a Kintchine's theorem associate with the variance of the increments a spectral measure. In this talk a formula for this spectral measure will be introduced. Other interesting fields with stationary increments will be introduced if I have enough time...
Tuesday 4 November
Sylvain Rubenthaler (Nice)
Title of the talk: Tree based functional expansions for particle models.
Abstract: We are interested in particle systems, or one could say equivalently "empirical measures", used to approximate various measures, solution of complicated equations. The propagation of chaos property is the property common to all these systems that the law of q particles will become the law of q independent particles having the exact target law when then number of particles goes to infinity. The development of the error in the propagation of chaos leads to the use of trees to represent empirical measures. I will focus on Feynman-Kac models. In the first part of the talk, I will define these models and explicit the associated particle systems. In the second part, I will talk of the development of the error in the propagation and chaos and show what are the combinatorics tools involved. In the third part, I will explain why this propagation of chaos is central in particle systems and show which results can be derived from there. This talk might be of interest for people of the statistics department and of the mathematics department and also for graduate students.
Tuesday 11 November
Paul Malliavin (Paris)
Title of the talk: Energy dissipation towards higher modes : Euler hydrodynamics, Virasoro unitarizing measures.
Abstract: It s shown that incompressible fluid dynamic with a vanishing viscosity is not ergodic (Joint Work with A.B. Cruzeiro JFA (258) April 2008, page 1903-1925).
Tuesday 25 November
John Hosking (Imperial)
Title of the talk: A weak integration-by-parts formula for a Malliavin calculus of pure jump Lévy functionals
Abstract: A key result in the Malliavin calculus of Wiener functionals is a certain integration-by-parts (IBP) formula, which can, for example, be used to prove the existence of a regular density function for the law of certain non-degenerate Wiener functionals. Using the Picard [1996] approach to a Malliavin calculus of pure jump Lévy functionals (PJLFs) we show how a weak form of IBP formula can be constructed in that setting. We discuss the question of whether this weak IBP formula can be used to prove the existence of a regular density function for the law of certain non-degenerate PJLFs. This question remains an open one, and we indicate the difficulties or faults of some approaches to the problem.
Tuesday 2 December
Vlad Bally
Title of the talk: Tubes estimates for Itô processes.
Abstract: We consider a stochastic equation with path dependent coefficients dXt=s(t,ω, Xt)dWt+b(t,ω, Xt)dt and we denote τ=inf{t:| Xt - xt |>r}, where xt is a deterministic differentiable curve and r>0. Our aim is to give lower bounds for P(τ >T), that is, for the probability that Xt remains in a tube of radius r around the curve xt up to time T. We specify this result in two significant frameworks. First we consider an elliptic type framework, that is: the coefficients are globally bounded and globally Lipschitz continuous and s s*(t,ω, Xt)³λ>0. In this case we find out Gaussian type lower bounds for P(τ >T). Next we consider a log-normal type framework, that is dXt=s(t,ω)XtdWt+b(t,ω)Xtdt with s and b bounded and s s*³λ>0. In this case the lower bounds are of log-normal type.
Finally we give some applications of our result. They are two cases: If we assume sufficient regularity for the coefficients s and b then the law of Xtis absolutely continuous and we obtain lower bounds for the density. But if we have less regularity for the coefficients, then we are not able to estimate the density. Nevertheless we are able to give lower bounds for E(f(Xt)) for a large class of functions f and this give lower bounds for the price of European options in finance. Moreover we give lower bounds for the price of Asian options.