Stochastic Analysis Seminar

The seminar meets during the term on Tuesdays Department of Mathematics, Huxley Building. All are welcome to attend the meetings.

Spring Term Program, 2012

Date/time: Tuesday 17 January (3pm)

Location: Room 139

Mike Giles (Oxford University)

Title of the talk: Multilevel Monte Carlo methods

Abstract: In the last 5 years there has been a growing amount of research on multilevel Monte Carlo methods by a number of groups.   In this talk I will give an overview of this work, with a particular emphasis on:

· the simplicity of the approach

· its applicability to a wide range of problems

· the scope for creativity in designing particularly efficient multilevel algorithms

· item progress in the numerical analysis of multilevel algorithms

Applications which will be discussed include: SDEs driven by Brownian motion, jump-diffusion and Levy processes, SPDEs and stochastic models for chemical reactions.

The talk is based on research with a number of collaborators, as well as research by others.  Further information on multilevel Monte Carlo research is available from: http://people.maths.ox.ac.uk/gilesm/mlmc_community.html

Date/time: Tuesday 24 January (3pm)

Location: Room 139

Albert N. Shiryaev (Moscow State University)

Title of the talk: Around the proof of the criteria for the uniform integrability of Brownian stochastic exponentials

Abstract: The talk deals with various sufficient conditions and their proofs for the uniform integrability of the exponential martingales of the form

E[λ]_t=exp(λB_{min(t, τ )}-1/2min(t, τ)), t≥0,

where B is a Brownian motion and τ is a stopping time. We show, for example, that the Novikov criterion E[e ^λ2τ/2]<∞ can be obtained from the criterion (Liptser, Shiryaev) E[e ^{λ2(1+ε)τ/2}]<∞, ε>0, whose proof uses only the Hölder inequality. Also we discuss the criterion of the type E[e ^{λ2(τ-φ(t))τ/2}]<∞ for some classes of the functions φ=φ(s).

Date/time: Thursday 26 January (5pm) Please note different time and venue

Location: 308

Mathematics Department Colloquium

Wendelin Werner (ORSAY)

Title of the talk: Random surfaces, random geometries

Abstract: Is there a natural universal analogue of Brownian motion for random surfaces or membranes?
If so, are there essential differences between these random real-valued functions defined on an interval (i.e. Brownian motion) or on a piece of the plane (this random membrane)? Are they really random surfaces? Is there a way to describe their topography?
We will try to discuss and partially answer such questions.

Date/time: Tuesday 31 January (3pm)

Location: Room 139

Nikolas Kantas (Imperial College London)

Title of the talk: Linear variance bounds for particle approximations of discrete time homogeneous Feynman-Kac formulae

Abstract: We are primarily interested in the non-asymptotic variance of particle approximations of discrete time homogeneous Feynman-Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance, rare event estimation and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman-Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. These are weaker conditions than the ones appearing in previous seminal works of P. Del Moral and his co-authors. We will illustrate with examples that our conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state spaces: 1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and 2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition. If time permits we will discuss briefly also the relevance of the conditions used and the results in the context of risk sensitive control. 

This talk is on joint work with Nick Whiteley and Ajay Jasra.  

Date/time: Tuesday 7 February (3pm)

Location: Room 139

Samuel Cohen (University of Oxford)

Title of the talk: Uncertainty and nonlinear expectations

Abstract: Decision making in the presence of uncertainty is a mathematically delicate topic. In this talk, we consider coherent sublinear expectations on a measurable space, without assuming the existence of a dominating probability measure. By considering discrete-time `martingale' processes, we show that the classical results of martingale convergence and the up/downcrossing inqualities hold in a `quasi-sure' sense. We also give conditions, for a general filtration, under which an `aggregation' property holds, generalising an approach of Soner, Touzi and Zhang (2011). From this, we extend various results on the representation of conditional sublinear expectations to general filtrations under uncertainty.

Date/time: Tuesday 14 February (3pm)

Location: Room 139

Xuerong Mao (University of Strathclyde)

Title of the talk: Stationary Distribution of Stochastic Population Systems

Abstract: In this talk we consider the stochastic  differential equation (SDE) population model $$dx(t) = \diag(x_1(t), \cdots, x_n(t)) [(b + Ax(t))dt +\sigma dB(t)]$$
for n interacting species. The main aim is to study the stationary distribution of the solution. It is known if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is   to find out what happens if the noise is relatively small.  In this talk we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution.  Computer simulations will
be used to illustrate the theory.

Date/time: Tuesday 21 February (3pm)

Location: Room 139

David Hobson (University of Warwick)

Title of the talk: The Skorokhod embeddings problem: some new uses for some old embeddings

Abstract: The Skorokhod embedding problem (SEP) for Brownian motion W is, given a centred probability measure m, to find a stopping time T such that the stopped process W_T has law m. Azema and Yor and later Perkins (and many others) gave explicit solutions to the SEP with particular optimality properties.

The model independent pricing problem, is given the prices of vanilla options but under no further assumptions on the model, to give model independent prices and hedges for co-maturing exotic options.

In this talk we show that the Azema and Yor and Perkins have some, perhaps surprising, additional optimality properties. Further we discuss the link between the SEP and the model independent pricing problem and show how the Perkins embeddings can be used to give bounds on the on the prices of variance swaps.

Date/time: Tuesday 28 February (3pm)

Location: Room 139

Emmanuel Gobet (Ecole Polytechnique)

Title of the talk: Almost sure optimal stopping times

Abstract: In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error in a multidimensional Itô model when the discrete rebalancing dates are stopping times. We investigate the convergence, in an almost sure sense, of the renormalized quadratic variation of the hedging error, for which we exhibit an asymptotic lower bound for a large class of stopping time strategies. Moreover, we make explicit a strategy which asymptotically attains this lower bound a.s. . Remarkably, the results hold under great generality on the payoff and the model. Our analysis relies on new results enabling to control a.s. processes, stochastic integrals and related increments. Extensions to transaction costs are also discussed.

Date/time: Tuesday 6 March (3pm)

Location: Room 139

José Manuel Corcuera (Universitat de Barcelona)

Title of the talk: Ambit processes and related issues

Abstract: : Ambit processes and related issues

Abstract: Ambit processes are processes of the form  

                Y(x)=∫_{A(x)}g(x-ξ)σ(ξ)W(dξ),

where x∈Rⁿ,  g:Rⁿ→R, with g(x₁,..,x_{n})=0 if x₁<0 (the first coordinate indicates time) , σ is the intermittency or volatility parameter and A(x) is the so called “ambit set”. In this talk we will review

some applications of ambit processes in turbulence and finance  and some mathematical problems connected with them.

Date/time: Tuesday 13 March (3pm)

Location: Room 139

Nadia Sidorova (University College London)

Title of the talk: Localisation and ageing in the Parabolic Anderson model

Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the transport through a random field of sinks and sources and is actively studied in mathematical physics.

We discuss, for a class of i.i.d. potentials, the intermittency effect occurring for such potentials, which manifests itself in increasing localisation and randomisation of the solution. We also discuss the ageing behaviour of the model as the time increases.

Autumn Term Program, 2011

Date/time: Tuesday 11 October (3pm)

Location: Room 139

Lucian Beznea (Institute of Mathematics, Romanian Academy)

Title of the talk: The semigroup approach for measure-valued branching processes and a nonlinear Dirichlet problem

Abstract: We use a branching Markov process on the space of finite configurations to solve a Dirichlet problem associated with the operator Δu+u². We follow the pioneering works of M. Nagasawa, N. Ikeda, S. Watanabe, M.L. Silverstein, and the approach of E.B. Dynkin.

Date/time: Tuesday 25 October (3pm)

Location: Room 139

Antoine Jacquier (Imperial College London)

Title of the talk: Generalised small-noise expansion for projected diffusions and applications

Abstract: Given a diffusion in Rⁿ, we prove a small-noise expansion for the density of its projection on a subspace of dimension p (p≤n). Our proof relies on the Laplace method on Wiener space and stochastic Taylor expansions in the spirit of Azencott-Benarous-Bismut. Our result (assuming Hormander's condition on the vector fields) applies both (i) to small-time asymptotics and (ii) to the tails of the distribution. In the context of stochastic volatility models, we recover the Busca-Berestycki-Florent formula (applying (i)) and Gulisashvili-Stein expansion (from (ii)).

This is a joint work with J.D. Deuschel (TU Berlin), P. Friz (TU Berlin) and S. Violante (Imperial College London).

Date/time: Tuesday 1 November (3pm)

Location: Room 139

Umut Cetin (London School of Economics)

Title of the talk: Explicit construction of a dynamic Bessel bridge of dimension 3

Abstract: Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V(t) satisfies V(t) > t for all t>0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V(T), T is the first time that Z hits 0. We also provide the semi-martingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V(T). We call this a dynamic Bessel bridge since V(T) is not known in advance. Our study is motivated by insider trading models with default risk.

Date/time: Tuesday 8 November (3pm)

Location: Room 139

Internal Stochastic Analysis meeting (speakers: Cass, Crisan, Ottobre, Ortiz-Latorre, Veraart)

Date/time: Tuesday 15 November (3pm)

Location: Room 139

Desmond Higham (University of Strathclyde)

Title of the talk: Multi-level Monte Carlo for Stochastic Chemical Kinetics

Abstract: Mike Giles (Multilevel Monte Carlo path simulation, Operations Research 56(3):607-617, 2008) devised and analysed a remarkable new algorithm that dramatically improves the computational complexity of Monte Carlo simulation for stochastic differential equations. We show how to extend these ideas to the case of a continuous time, discrete space, Markov chain, focussing on the setting of stochastically modelled chemical kinetics. The extension involves a non-trivial coupling device and also exploits the existence of exact (but expensive) simulation algorithms to produce an unbiased estimator. By analysing the new algorithm under an appropriate scaling, we show that popular Gillespie/next reaction method/tau-leaping approaches can be dramatically improved in a manner that we quantify precisely. Computational results will be given.

This is joint work with David Anderson (Wisconsin).

Date/time: Tuesday 22 November (3pm)

Location: Room 139

Francois Delarue (University of Nice)

Title of the talk: Unique solvability of singular differential equations driven by degenerate noise.

Abstract: We here discuss examples of differential equations driven by singular coefficients and by a degenerate noise for which strong existence and uniqueness hold. The first example is a stochastic Hamiltonian system. Existence and uniqueness are established for Holder continuous drifts by taking benefit of the regularization property of the underlying operator. The second example is of Poisson-Vlasov type. Strong solvability is discussed for a system of N interacting particles whose velocities are driven by a discontinuous forcing of mean-field type and by a noise that degenerates along the diagonal.

It is then proven that noise restores existence and uniqueness of a solution and prevents coalescence of particles for any initial condition outside the diagonal. The proof relies on the hypoellipticity of the particle system away from the diagonal.

This is joint work with F. Flandoli (Pisa), D. Vincenzi (Nice) and P.E. Chaudru de Raynal (Nice).

Date/time: Tuesday 29 November (3pm)

Location: Room 139

Michela Ottobre (Imperial College London)

Title of the talk: Markovian approximations of classical open systems

Abstract: We consider a class of hypoelliptic Markovian approximations to the non Markovian Generalized Langevin Equation (GLE). We present several properties of such approximations: ergodicity, exponentially fast decay to equilibrium (hypocoercivity), sharp estimates on the short time behaviour of the n-th derivative of the semigroup.

We shall also discuss a technique to determine the exact rate of exponential  convergence to equilibrium and how the hypoelliptic properties of the generator are related to the ergodicity of the process.

Date/time: Thursday 1 December (5pm)

Location: TBA

Mathematics Department Colloquium

Paul Embrechts (ETH)

Title of the talk: The Financial Crisis as a Crisis of Financial Mathematics

Abstract: Around the 2007-09 subprime crisis, mathematicians/financial engineers were partly blamed for what went wrong during the crisis. I will first give a critical overview of these accusations followed by some examples showing that mathematics has an important role to play going forward. I will also make recommendations for teaching as well as research.

Date/time: Tuesday 6 December (3pm)

Location: Room 139

Joint Stochastic Analysis/Statistics seminar

Eric Moulines (Institut Télécom / Télécom ParisTech (ENST))

Title of the talk: Long-term stability of sequential Monte Carlo methods under verifiable conditions

Abstract: The theory of particle filtering is well investigated and there is number of available convergence results concerning, e.g. Lp error bounds and weak convergence -- see the monographs Del Moral (2004) and Bain and Crisan (2008) for extensive overviews of recent theoretical developments. Most of these results establish the convergence, as the number of particles N tends to infinity, of the particle filter for a fixed time step n. In standard theoretical motivations of the particle filter, each recursive importance sampling update (selection followed by mutation) of the particle cloud is based on the implicit assumption that the ancestor sample approximates perfectly well the predictor at the previous time step; however, since also the ancestor sample is obtained through importance sampling, one could expect that the errors induced at the different updating steps accumulate and, consequently, that the total error propagated through the algorithm increases with n. This would make the algorithm useless in practice.

Fortunately, it has been observed empirically by several authors that the convergence of particle filters appears to be uniform in time also for very general HMMs. Nevertheless, even though long-term stability is essential for the applicability of particle filters, most existing time uniform convergence results are obtained under assumptions that are generally not met in practice. The aim of this talk is thus to establish novel such results under mild and easy verifiable assumptions.

Date/time: Friday 9 December (1pm) Please not different time and venue

Location: Room 130

Sebastian Reich (Universität Potsdam)

Title of the talk: Ensemble transform filters for nonlinear dynamical systems

Abstract: Given a nonlinear dynamical systems with chaotic solution behaviour, we consider tracking a (unknown) reference solution using partial observations of the system. The talks will focus on sequential filtering/data assimilation algorithms based on ensembles of stochastic particles. Building on the popularity of the ensemble Kalman filter as well as its known limitations, I will discuss other variants of ensemble transform filters which do not rely on a Gaussian approximation to the ensemble of particles.

Summer Term Program, 2011

Date/time: Tuesday 3 May (3pm)

Location: Room 140

Arturo Kohatsu Higa (Ritsumeikan University)

Title of the talk: Methods to Deal with Non-smooth Coefficients in Malliavin Calculus

Abstract: Until recently it was thought that Malliavin Calculus is a tool to be used with stochastic equations (e.g diffusions) with smooth coefficients under hypoelliptic conditions. The method was general enough so that it could be used for variety of other equations without much problem. On the other hand, there are refined analytical techniques to prove the existence of fundamental solutions for elliptic diffusions under almost no regularity conditions. This has (and still remains) remained the difference in both methods for a long time. The recent efforts in the area are to reduce the smoothness requirements when applying Malliavin Calculus to stochastic equations.

We present a general method that allows to use Malliavin Calculus for stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô -Taylor expansion in order to obtain regularity properties for the density of a hypoelliptic additive type random variable at a fixed time. We will show the methodology to the case of the Lebesgue integral of a diffusion. This is joint work with A. Tanaka.

Time permitting we may show other extensions that we are currently developing.

Date/time: Tuesday 17 May (3pm)

Location: Room 139

Tom Kurtz (University of Wisconsin-Madison)

Title of the talk: Poisson representations of branching Markov and measure-valued branching processes

Abstract: Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a “level'', but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov processes, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0, r]. For the limiting measure-valued process, at each time t, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure the product of Lebesgue measure and the state of the desired measure-valued process. The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris's convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

Date/time: Tuesday 21 June (3pm)

Location: 340

Boris Rozovsky (Brown University)

Title of the talk: On Quantized Stochastic Navier-Stokes Equation

Abstract: A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term $u{\nabla}u$. This perturbation is unbiased in that the expectation of a solution of the perturbed/quantized equation solves the deterministic Navier-Stokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. The generalized solution can be obtained as a limit or an inverse of solutions to corresponding quantized equations. It is shown that the generalized solution is a Markov process.

Joint work with R. Mikulevicius

Date/time: Tuesday 28 June (3pm)

Location: 340

Jie Xiong (University of Tennessee)

Title of the talk: Large deviation principle for diffusion processes under a sublinear expectation

Abstract: We represent the exponential moment of the Brownian functionals under a nonlinear expectation according to the solution to a backward stochastic differential equation. As an application, we establish a large deviation principle of the Freidlin and Wentzell type under the corresponding nonlinear probability for diffusion processes with a small diffusion coefficient.

This talk is based on a joint paper with Z.J. Chen.

Spring Term Program, 2011

Date/time: Tuesday 18 January (3pm)

Location: Room 139

Francois Delarue (University of Nice)

Title of the talk: Singular FBSDEs and Emissions Derivatives

Abstract: We here investigate a class of forward-backward SDEs arising in the analysis of emissions markets. The resulting FBSDEs at hand are doubly singular: the noise is degenerate and the payoff function is of Heaviside type. We then show that the model has some analogy with a partially degenerate equation of conservation law. Taking benefit of this analogy, we establish conditions under which both existence and uniqueness hold in a suitable sense. We also show that the singularities of the equation conspire to produce a non-trivial mass point at maturity: on the emissions market we consider, the price of the emission allowance is not Markov for emission scenarios that end at the singular point of the payoff function.

Date/time: Tuesday 25 January (3pm)

Location: Room 139

Salvador Ortiz-Latorre (Imperial College)

Title of the talk: Weak convergence of nonlinear functionals of Gaussian processes and Malliavin calculus

Abstract: In this talk we will introduce a new characterization of the weak convergence of a sequence of multiple stochastic integrals to the normal law. This new characterization is in terms of the Malliavin derivative of the elements of the sequence, giving a novel application of Malliavin calculus. We will also discuss the multidimensional version of this result as well as some applications and partial extensions.

Date/time: Wednesday 2 February (3pm) NOTE DIFFERENT DAY OF THE WEEK AND DIFFERENT ROOM

Location: Room 130

Jean-Francois Chassagneux (Evry University)

Title of the talk: Doubly Reflected BSDEs with Call Protection and their Approximation

Abstract: We study the numerical approximation of doubly reflected backward stochastic differential equations with intermittent upper barrier (RIBSDE) i.e. reflected BSDEs in which the upper barrier is only active on certain random time intervals. From the point of view of financial interpretation, RIBSDEs arise as pricing equations of game options with call protection, in which the call times of the option’s issuer are subject to constraints preventing the issuer from calling the option on certain random time intervals.

This is a joint work with S. Crépey (Université d'Evry-Val d'Essonne).

Date/time: Tuesday 8 February (3pm)

Location: Room 139

Pierre Del Moral (INRIA, Bordeaux)

Title of the talk: On the Approximations of Multiple target filtering equations

Abstract: We consider the problem of estimating a latent point process, given the realization of another point process on abstract measurable state spaces. First, we establish an expression of the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning and augmentation with extra points. We present an original analysis based on a self-contained random measure theoretic approach combined with reversed Markov kernel techniques. In the second part, we analyse the exponential stability properties of nonlinear multi-target filtering equations. We prove uniform convergence properties 

w.r.t. the time parameter of a rather general class of stochastic filtering algorithms, including sequential Monte Carlo type models and mean field particle interpretation models. We illustrate these results in the context of the Bernoulli and the Probability Hypothesis Density filter.

Date/time: Tuesday 15 February (3pm)

Location: Room 139

Marta Sanz Solé (University of Barcelona)

Title of the talk: Hitting probabilities for systems of non-linear stochastic wave equations in spatial dimension k∈{1, 2, 3}

Abstract: We consider a system of d non-linear stochastic wave equations in spatial dimension k∈{1, 2, 3}. The driving noise is d-dimensional, white in time and with a spatially homogeneous covariance defined by a Riesz kernel with exponent β∈(0,2⋀k). We establish an upper bound on hitting probabilities of the solution to the system in terms of Hausdorff measure of dimension arbitrarily close (but smaller) to d-2(k+1)/(2- β). This uses properties on the density of the solution and L^p estimates of increments of the solution at two different points. Then, we prove upper bounds for the L^p moments of the inverse of the eigenvalues of the Malliavin covariance matrix of the R^2d -valued random vector (u(s,y), u(t,x)). In particular, for small eigenvalues, we quantify the rate of explosion as (s,y) → (t,x). This yields upper bounds for the join density of (u(s,y), u(t,x)) and eventually, lower bounds on hitting probabilities in terms of the Newtonian capacity of dimension arbitrarily close to (but bigger than) d+d²/(2- β) -2(k+1)/(2- β). This is joint work with R.C. Dalang.

Date/time: Tuesday 1 March (3pm)

Location: Room 139

Nizar Touzi (Ecole Polytechnique)

Title of the talk: Model independent bound for option pricing: a stochastic control approach

Abstract: We develop a stochastic control approach for the derivation of model independent bounds for derivatives under various calibration constraints. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples.

Date/time: Tuesday 8 March (3pm)

Location: Room 139

Xu Mingyu (Institute of Applied Mathematics, Beijing)

Title of the talk: Reflected  BSDE with a constraint and its application

Abstract: Non-linear backward stochastic differential equations (BSDEs in short) were firstly introduced by Pardoux and Peng (1990), who proved the existence and uniqueness of the adapted solution, under smooth square integrability assumptions on the coefficient and the terminal condition, and when the coefficient g(t,ω,y,z) is Lipschitz in (y,z) uniformly in (t, ω). From then on, the theory of backward stochastic differential equations (BSDE) has been widely and rapidly developed. And many problems in mathematical finance can be treated as BSDEs. The natural connection between BSDE and partial differential equations (PDE) of parabolic and elliptic types is also important applications. In this talk, we study a new development of BSDE, BSDE with constraint and reflecting barrier.

The existence and uniqueness results are presented and we will give some application of this kind of BSDE.

Date/time: Thursday 10 March (3pm) (Analysis Seminar - NOTE DIFFERENT DAY OF THE WEEK)

Location: Room 139

Erwin Bolthausen (University of Zurich)

Title of the talk: Non-ballistic random walks in random environments

Abstract: We review a multiscale method for analysing the exit distributions from large sets for random walks in random environments in dimensions three and above. We also present work in progress on the same problem in the critical dimension two. (Joint work with Ofer Zeitouni).

Autumn Term Program, 2010

Date/time: Tuesday 12 October (3pm)

Location: Room 139

Jie Xiong (University of Tennessee)

Title of the talk: Super Brownian Motion as the unique strong solution of an SPDE

Abstract: A stochastic partial differential equation (SPDE) wii be derived for the super Brownian motion regarded as a distribution function valued process. The strong uniqueness for the solution to this SPDE is obtained by a connection between SPDEs and backward doubly stochastic differential equations. Similar results for the Fleming-Viot process will also be presented.

Date/time: Tuesday 19 October (4.10pm) (Please note different time and location)

Location: Room 130

Joint AMMP Colloquium – Stochastic Analysis seminar

Georg Gottwald (University of Sydney)

Title of the talk: A variance constraining ensemble Kalman filter: How to improve forecast using climatic data of unobserved variables

Abstract: Data assimilation aims to solve one of the fundamental problems of numerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system. A standard tool in attacking this filtering problem is the Kalman filter.

Date/time: Tuesday 26 October (3pm)

Location: Room 139

Martijn Pistorius (Imperial College London)

Title of the talk: On inverse-first passage problems arising in credit risk

Abstract: An inverse first passage problem is to find a boundary such that the first-passage time follows a given distribution. This problem is of interest in credit risk valuation. We will discuss tractable solutions to this problem, and present as application the computation of the credit valuation adjustment for a swap. This is joint work with Mark Davis.

Date/time: Tuesday 2 November (3pm)

Location: Room 139

Francois Delarue (University of Nice)

Title of the talk: Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs

Abstract: We investigate the Holder regularity of the solution of a possibly degenerate elliptic PDE of nonlinear type. The diffusion coefficient of the PDE may degenerate when the gradient of the solution is small. A typical example is given by the p-Laplace equation. The proof is purely probabilistic and relies on a variant of the Krylov and Safonov argument that applies in the non-degenerate and linear framework. In a word, the point is to introduce a suitable representation process that visits the surrounding space sufficiently.

Date/time: Tuesday 9 November (4.10pm) (Please note different time and location)

Location: Room 130

Joint AMMP Colloquium – Stochastic Analysis seminar

Andrew Stuart (University of Warwick)

Title of the talk: Connections Between (S)PDEs and MCMC in a Hilbert Space

Abstract: TBA.

Date/time: Tuesday 16 November (3pm)

Location: Room 139

Jean-Francois Chassagneux (Evry University)

Title of the talk: A discrete-time approximation for reflected BSDEs related to ``switching problem''

Abstract: We present a discrete-time approximation for a class of multi-dimensional obliquely reflected BSDEs. In the 2-dimensional case, they were introduced by Hamadene and Jeanblanc and latter generalized by Hu and Tang and Hamadene and Zhang. They are closely related to ``switching problem'', encountered in real option pricing e.g.. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver of the BSDEs does not depend on the variable Z, the error on the approximation grid points between the solution and the numerical scheme is of order 1/2-e, e>0.

Date/time: Tuesday 23 November (3pm)

Location: Room 139

Joscha Diehl (TU Berlin)

Title of the talk: BSDEs with rough drivers

Abstract: Classically, the driver of a backward stochastic di_erential equation (BSDE) is a Lipschitz continuous function that is integrated with respect to Lebesgue measure. We introduce a new class of BSDEs where the driver consists of an additional integral that can posses much less regularity in time. We show existence, uniqueness and stability results for these equations and establish the connection to backward doubly stochastic di_erential equations. Our interest in these equations has been partly motivated by their connection to PDEs driven by rough paths, for which we establish a Feynman-Kac type formula. This is

joint work with Peter Friz (TU Berlin). No prior knowledge of rough path theory will be essential to follow the talk.

Date/time: Tuesday 14 December (3pm)

Location: Room 139

Greg Pavliotis (Imperial College London)

Title of the talk: Analytical and numerical methods for SPDEs with multiple scales

Abstract: In this talk we will present analytical and numerical techniques for studying stochastic partial differential equations with multiple scales. After showing a rigorous homogenization theorem for SPDEs with quadratic nonlinearities, we present a numerical method for solving efficiently SPDEs with multiple scales. We then apply these analytical and numerical techniques to several examples, including the stochastic Burgers and the stochastic Kuramoto-Shivashinsky equation. This is joint work with D. Blomker and M. Hairer (analysis) and with A. Abdulle (numerical analysis).

Summer Term Program, 2010

Date/time: Tuesday 27 April (4.10pm) NOTE DIFFERENT TIME !

Location: Room 139

Marco Romito

Title of the talk: Analysis of a model for amorphous surface growth

Abstract: We consider a semilinear fourth order equation arising in surface growth caused by epitaxy or sputtering. In the first part of the talk, we give a complete analysis of the one dimensional problem forced by space-time white noise in the framework of Markov solutions. In the second part we analyse the unforced case and give conditions for the emergence of blow up. Finally we briefly introduce the two dimensional problem, which corresponds to the physical case, and give a few preliminary existence results.

Date/time: Tuesday 4 May (3pm) Reading group on Stochastic Differential Equations on Manifolds

Location: Room 139

Jason Lotay

Title of the talk: Frame Bundles and Connections

Date/time: Tuesday 11 May (3pm)

Location: Room 139

Eulalia Nualart (Paris 13)

Title of the talk: Strict positivity and lower Gaussian bounds for the density

of a class of spatially homogeneous SPDEs

Abstract: We consider the following class of spatially homogeneous SPDEs

\[Lu(t,x)=\sigma(u(t,x)) \dot{W}(t,x)+b(u(t,x)), \; \; t \in ]0,T], x \in\r^d, \] where $L$ is a second order differential operator, $\sigma,b$ are Lipschitz functions and $W$ is a Gaussian noise which is white in time and has a spatially homogeneous covariance.

We will start recalling known results on existence, uniqueness, and existence and smoothness of the density for the solution of this class of SPDEs under sufficient conditions on the fundamental solution of the deterministic equation $Lu=0$. We will then give sufficient conditions to obtain the strict positivity of the density. In the case where $L$ is the heat operator we will prove a Gaussian type lower bound for the density.

These results are obtained using techniques of Malliavin calculus. The motivation for studying the strict positivity of the density is to develop in a further work potential theory for this class of SPDEs. We will recall some of the ongoing works.

Date/time: Tuesday 18 May (4pm) Note the later time!

Location: Room 139

Laszlo Gyorfi

Title of the talk: Portfolio games

Abstract: The growth optimal empirical portfolio selection rules are based on three

sources:

- rebalancing,

- nonparametric estimates,

- machine learning algorithm for aggregation.

In this seminar I illustrate the rebalancing via examples: Kelly games, horse racing, St.Petersburg games.

Spring Term Program, 2010

Tuesday  26 January 

Bruno Bouchard

Title of the talk: Optimal Control under Stochastic Target Constraints

Abstract: We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s. constraint $Z^\nu(T)\in G\subset\R^{d+1}$ $\P a.s.$ at some final time $T>0$.  When the set is of the form $G:=\{(x,y)\in \R^d\x \R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $ \nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE.  Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $\Esp{g(Z^\nu(T))}\ge 0$ or $\Pro{g(Z^\nu(T))\ge 0}\ge p$. 

Tuesday  9 February 

Anis Matoussi

Title of the talk: The obstacle problem for quasilinear stochastic PDE's and the probabilistic interpretation of the solution via BSDE's and regular potentials

Abstract: We prove an existence and uniqueness result for the obstacle problem of quasilinear parabolic stochastic PDEs. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation. Moreover, we examine also the potential and the measure associated to a continuous increasing process. We call such potentials and measures, regular potentials, respectively regular measures.

This is a joint work with L. Stoica (University of Bucharest), which will appear in the Annals of Probability.

Tuesday  23 February 

Peter Imkeller

Title of the talk: Utility indifference hedging, BSDE of quadratic growth and measure solutions

Abstract: A financial market model is considered on which agents (e.g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. They are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indifference. This hedging concept is interpreted by means of martingale optimality, and solved with BSDE with drivers of quadratic growth in the control variable. We investigate a new concept of solutions for BSDE of this type, which we call measure solutions and which corresponds to the concept of risk-neutral measures in arbitrage theory. We show that strong solutions of BSDE induce measure solutions, and present an algorithm by which measure solutions can be constructed without reference to strong ones. It yields solutions in new scenarios. For the case of unbounded terminal conditions existence and uniqueness questions become very difficult. We illustrate a wealth of different scenarios by giving examples and counterexamples.

This is joint work with S. Ankirchner, A. Fromm, G. Heyne, Y. Hu, M. Muller, A. Popier, J. Zhang.

Tuesday  2 March

Tom Kurtz (University of Wisconsin-Madison)

Title of the talk: Identifying separated time scales in stochastic models of reaction networks

Abstract. For chemical reaction networks in biological cells, reaction rates and chemical species numbers may vary over several orders of magnitude. Combined, these large variations can lead to subnetworks operating on very different time scales. Separation of time scales has been exploited in many contexts as a basis for reducing the complexity of dynamic models, but the interaction of the rate constants and the species numbers makes identifying the appropriate time scales tricky at best. Some systematic approaches to this identification will be discussed and illustrated by application to one or more complex reaction network models.

Tuesday  16 March 

Umut Cetin (LSE)

Title of the talk: On Dynamic Markov Bridges

Abstract.  Motivated by the insider trading models of Kyle and Back, we present a theory of Markov bridges. We call them 'dynamic' since the terminal value is not known in advance. In this talk I will briefly describe how to construct a diffusion which is a martingale and whose terminal value is defined by the terminal value of another martingale diffusion observed continuously in time. Then, I will discuss the construction of a Brownian motion who is conditioned to hit 0 for the first time at a given function of the hitting time of another Gaussian martingale. Our approach is based on nonlinear filtering theory and parabolic PDEs. In particular, we obtain a remarkable PDE whose solution gives the solution to the associated nonlinear filtering problem.

The talk is based on joint works with L. Campi and A. Danilova.

Additional talks

Jonathan Mattingly - Monday, 15 March, 1-2pm, Imperial College, Room 139

Dan Stroock - Thursday, 25 March, 4.30-5.30pm, Imperial College, Room 139

Autumn term Program, 2009

Tuesday  6 October 

Arnaud Doucet

Title of the talk: Forward Smoothing using Sequential Monte Carlo with Application to Recursive Parameter Estimation

Abstract: Sequential Monte Carlo (SMC) methods are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. We propose a new SMC algorithm to compute the expectation of additive functionals recursively. Compared to the standard path space SMC estimator whose asymptotic variance increases quadratically with time even under favourable mixing assumptions, the asymptotic variance of the proposed SMC estimator only increases linearly with time. We show how this allows us to perform recursive parameter estimation using SMC algorithms which do not suffer from the particle path degeneracy problem.

Joint work with P. Del Moral (INRIA Bordeaux) and  S.S. Singh (Cambridge University)

Tuesday  13 October

Alice Guionnett

Title of the talk: The single ring theory

Abstract: We study the spectrum of some general ensembles of non-normal random matrices and show that its empirical measure converges to a deterministic measure whose support is a single ring. This is surprising as the empirical measure of the singular values of these ensembles can be as disconnected as we wished. Our result generalizes a work by Feinberg and Zee 97'.

JW with M. Krishnapur and O. Zeitouni

Tuesday 20 October (4.10pm) Room 139 (Please note different time/room)

Joint AMMP Colloquium – Stochastic Analysis seminar

Jochen Voss

Title of the talk: Finite Difference Approximations of SPDEs

Abstract: We study the problems which can occur when naively using finite difference approximations for SPDEs.  It transpires that sometimes the discretised equations converge to the "wrong" SPDE when the grid size goes to zero, the error manifests itself as an additional drift term in the limiting equation.  We illustrate how one can sometimes guess the form of this additional drift term.

Tuesday 3 November (4.10pm) Room 139 (Please note different time/room)

Joint AMMP Colloquium – Stochastic Analysis seminar

Grant Lythe

Title of the talk: Stochastic dynamics of kinks

Abstract: Localised coherent structures are a striking feature of noisy, nonlinear, spatially-extended systems. In one space dimension with local bistability, coherent structures are kinks.  At late times, a steady-state density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Long-term averages can be calculated using the transfer-integral method, developed in the 1970s, giving exact results that can be compared with large-scale numerical solutions of SPDE.  More recently, the equivalence between the stationary density (in space) of an SPDE and that of a suitably-chosen diffusion process (in time) has been used, by a different community of researchers, to perform sampling of bridge diffusions.  In this talk, diffusion-limited reaction is the name given to a reduced model of the SPDE dynamics, in which kinks are treated as point particles. Some quantities, such as the mean number of particles per unit length, can be calculated exactly.

Tuesday 10 November

Sylvain Rubenthaler

Title of the talk: Introduction to particle filters - a short course with proofs

Abstract: This is a very brief course on nonlinear filtering. I will define what is the nonlinear filter, prove various formulas for computing the optimal filter. I will then detail the definition a particle filter (which is the name for a special algorithm). Finally, I will prove the convergence of the particle filter towards the optimal filter (when the number of particles goes to infinity, in a very weak sense).

Tuesday 24 November

Tusheng Zhang

Title of the talk: Stochastic partial differential equations with reflection

Abstract: This work is concerned with white noise driven SPDEs with 

reflection.  The existence and uniqueness of the  solution will be  

discussed. Various properties of the solution  will be presented, for 

example,  the   strong Feller property, Harnack inequalities,   Varadhan type small time asymptotics  and also the large deviations.

Tuesday 1 December

David Elworthy

Title of the talk: Geometric approach to filtering: some infinite dimensional illustrations

Abstract: Suppose we have an SDE on which lies over an SDE on for the natural projection of  to . With some “cohesiveness" assumptions on the SDE on . we can decompose the SDE on the big space and so describe the conditional law of its solution given knowledge of its projection. The same holds for suitable SDE's on manifolds, and in some infinite dimensional examples arising from SPDE's and stochastic flows. This approach will be illustrated by considering the conditional law of solutions of a simple evolutionary SPDE given (a) the integral of the solution over the space variables and (b) the values of the solution at one point of space, and also by looking at the problem of conditioning a stochastic flow by knowledge of its one-point motion.

This is joint work taken from a monograph by myself, Yves LeJan, and

Xue-Mei Li, The Geometry of Filtering to appear in Birkhauser's “Frontiers

in Mathematics" series.

Tuesday 8 December

Huyên Pham

Title of the talk: Stochastic control under progressive enlargement of filtrations and applications to default risk management

Abstract: We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a backward induction. This general study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claim, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump component coming a priori from the default times. 

Tuesday 15 December Waldemar Hebish 

Title of the talk: Estimates for heat kernel on Lie groups

Abstract: We discuss long time pointwise and integral estimates for heat kernel and its gradient on solvable Lie groups.  We will present general analytic method and (for some specific groups) improvements using path integrals.

Wednesday  16 December    Pierre Del Moral  (Room 139) 

Title of the talk: An introduction to particle simulation of rare events

Spring term Program, 2009

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

Greg Pavliotis (Imperial)

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 local 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

Jerzy Zabczyk

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 talk 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 dX_t=s(t,ω, X_t)dW_t+b(t,ω, X_t)dt and we denote τ=inf{t:| X_t - x_t |>r}, where x_t 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 X_t remains in a tube of radius r around the curve x_t 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, X_t)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 dX_t=s(t,ω)X_tdW_t+b(t,ω)X_tdt 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 X_t is 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(X_t)) 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.

Past Seminars

Maps and Instructions:

· London Underground maps

· Getting to Imperial College. (do not go in the tunnel if you travel to South Kensington tube station.)

The simplest way to get here:

T                      Travel to the tube station Gloucester Road  (District, Circle, and Piccadilly Lines).  When you exit the station, turn left along Gloucester Road, crossing Cromwell Road 50 meters from the exit.  After 4-5 minutes walk along Gloucester Road, turn right to Queen's Gate Terrace. This is a short road leading directly to the entrance of the Huxley Building, at 180 Queen's Gate. We are on floor 6.

For additional information contact Dan Crisan.