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.

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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 sel f-contained random measure theoretic approach combined with reve rsed 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).

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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) (Imperial College London)
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
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).