Seminars 2009/10
Summer Term Program, 2010
Tuesday 27 April (4.10pm) NOTE DIFFERENT TIME !
Location: Room 139
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.
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
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
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
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 th e hitt ing time of another Gaussian martingale. Our approach is based on nonlinear filte ring 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 w i thout 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.
Abstract: TBA