Forthcoming events in this series


Mon, 11 Mar 2013

15:45 - 16:45
Oxford-Man Institute

Random FBSDEs: Burgers SPDEs, Rational Expectations / Consol Rate Models, Control for Large Investors, and Stochastic Viscosity Solutions.

NIKOLAOS ENGLEZOS
(University of Piraeus)
Abstract

Abstract: Burgers equation is a quasilinear partial differential equation (PDE), proposed in 1930's to model the evolution of turbulent fluid motion, which can be linearized to the heat equation via the celebrated Cole-Hopf transformation. In the first part of the talk, we study in detail general versions of stochastic Burgers equation with random coefficients, in both forward and backward sense. Concerning the former, the Cole-Hopf transformation still applies and we reduce a forward stochastic Burgers equation to a forward stochastic heat equation that can be treated in a “pathwise" manner. In case of deterministic coefficients, we obtain a probabilistic representation of the Cole-Hopf transformation by associating the backward Burgers equation with a system of forward-backward stochastic differential equations (FBSDEs). Returning to random coefficients, we exploit this representation in order to establish a stochastic version of the Cole-Hopf transformation. This generalized transformation allows us to find solutions to a backward stochastic Burgers equation through a backward stochastic heat equation, subject to additional constraints that reflect the presence of randomness in the coefficients. In both settings, forward and backward, stochastic Feynman-Kac formulae are derived for the solutions of the respective stochastic Burgers equations, as well. Finally, an application that illustrates the obtained results is presented to a pricing/hedging problem arising from mathematical finance.

In the second part of the talk, we study a class of stochastic saddlepoint systems, represented by fully coupled FBSDEs with infinite horizon, that gives rise to a continuous time rational expectations / consol rate model with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness and dependence on a parameter of adapted solutions. Making further the connection with quasilinear backward stochastic PDEs (BSPDEs), we are led to the notion of stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of a large investor is also provided as an application to this framework.

This is joint work with N. Frangos, X.- I. Kartala and A. N. Yannacopoulos*

Mon, 11 Mar 2013

14:15 - 15:15
Oxford-Man Institute

Pathwise approximation of SDE solutions using coupling

SANDIE DAVIE
(University of Edinburgh)
Abstract

The standard Taylor series approach to the higher-order approximation of vector SDEs requires simulation of iterated stochastic integrals, which is difficult. The talk will describe an approach using methods from optimal transport theory which avoid this difficulty in the case of non-degenerate diffusions, for which one can attain arbitrarily high order pathwise approximation in the Vaserstein 2-metric, using easily generated random variables.

Mon, 04 Mar 2013

15:45 - 16:45
Oxford-Man Institute

Uniformly Uniformly-Ergodic Markov Chains and applications

SAMUEL COHEN
(University of Oxford)
Abstract

If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.

Mon, 04 Mar 2013

14:15 - 15:15
Oxford-Man Institute

Bond Percolation on Isoradial Graphs

IOAN MANOLESCU
(University of Cambridge)
Abstract

The star-triangle transformation is used to obtain an equivalence extending over a set bond percolation models on isoradial graphs. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models, under some conditions. In particular this implies criticality for these models.

(joint with Geoffrey Grimmett)

Mon, 25 Feb 2013

15:45 - 16:45
Oxford-Man Institute

Nonnegative local martingales, Novikow's and Kazamaki's criteria, and the distribution of explosion times

JOHANNES RUF
(University of Oxford)
Abstract

I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure.

In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation

Mon, 25 Feb 2013

14:15 - 15:15
Oxford-Man Institute

Poisson random forests and coalescents in expanding populations.

SAM FINCH
(University of Copenhagen)
Abstract

Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one.

Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly.

We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D.

We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.

Mon, 18 Feb 2013

15:45 - 16:45
Oxford-Man Institute

A continuum of exponents for the rate of escape of random walks on groups

GIDI AMIR
(Bar-Ilan University)
Abstract

Abstract: A central question in the theory of random walks on groups is how symmetries of the underlying space gives rise to structure and rigidity of the random walks. For example, for nilpotent groups, it is known that random walks have diffusive behavior, namely that the rate of escape, defined as the expected distance of the walk from the identity satisfies E|Xn|~=n^{1/2}. On nonamenable groups, on the other hand we have E|Xn| ~= n. (~= meaning upto (multiplicative) constants )

In this work, for every 3/4 <= \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^\beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval.

Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions.(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. (Paper available at http://arxiv.org/abs/1203.6226)

No previous knowledge of random walks,automaton groups or wreath products is assumed.

Mon, 18 Feb 2013

14:15 - 15:15
Oxford-Man Institute

Rough paths, controlled distributions, and nonlinear SPDEs

NICOLAS PERKOWSKI
(Humboldt University, Berlin)
Abstract

Abstract: Hairer recently had the remarkable insight that Lyons' theory of rough paths can be used to make sense of nonlinear SPDEs that were previously ill-defined due to spatial irregularities. Since rough path theory deals with the integration of functions defined on the real line, the SPDEs studied by Hairer have a one-dimensional spatial index variable. I will show how to combine paraproducts, a notion from functional analysis, with ideas from the theory of controlled rough paths, in order to develop a formulation of rough path theory that works in any index dimension. As an application, I will present existence and uniqueness results for an SPDE with multidimensional spatial index set, for which previously it was not known how to describe solutions. No prior knowledge of rough paths or paraproducts is required for understanding the talk. This is joint work with Massimiliano Gubinelli and Peter Imkeller.

Mon, 11 Feb 2013

15:45 - 16:45
Oxford-Man Institute

Numerical Solution of FBSDEs Using a Recombined Cubature Method

Camilo Andres Garcia Trillos
(University of Nice Sophia-Antipolis)
Abstract

(Joint work with P.E. Chaudru de Raynal and F. Delarue)

Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity.

An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity.

In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm.

We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests.

References

[1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010.

[2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

Mon, 11 Feb 2013

14:15 - 15:15
Oxford-Man Institute

A randomluy forced Burgers equation on the real line

ERIC CATOR
(Delft University of Technology)
Abstract

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

Mon, 04 Feb 2013

14:15 - 15:15
Oxford-Man Institute

Filtration shrinkage, strict local martingales and the Follmer measure

MARTIN LARSSON
(EPFL Swiss Finance Institute)
Abstract

Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Mon, 28 Jan 2013

15:45 - 16:45
Oxford-Man Institute

Near-critical Ising mode.

CHRISTOPHE GARBAN
(ENS Lyon)
Abstract
In this talk, I will present two results on the behavior of the Ising model on the planar lattice near its critical point: (i) In the first result (joint work with F.Camia and C. Newman), we will fix the temperature to be the critical temperature T_c and we will vary the magnetic field h \geq 0. Our main result states that in the plane Z^2, the average magnetization at the origin behaves up to constants like h^{1/15}. This result is interesting since the classical computa- tion of the average magnetization by Onsager requires the external magnetic field h to be exactly 0 . (ii) In the second result (joint work with H. Duminil-Copin and G. Pete), we focus on the correlation length of the Ising model when h is now fixed to be zero and one varies instead the temperature T around T_c. In rough terms, if T
Mon, 28 Jan 2013

14:15 - 15:15
Oxford-Man Institute

Half planar random maps

OMER ANGEL
(University of British Colombia)
Abstract

Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

Mon, 21 Jan 2013

15:45 - 16:45
Oxford-Man Institute

The stochastic quasi-geostrophic equation

RONGCHAN ZHU
(Bielefeld University)
Abstract
In this talk we talk about the 2D stochastic quasi-geostrophic equation on T2 for general parameter _ 2 (0; 1) and multiplicative noise. We
prove the existence of martingale solutions and Markov selections for multiplicative noise for all _ 2 (0; 1) . In the subcritical case _ > 1=2, we prove existence and uniqueness of (probabilistically) strong solutions. We obtain the ergodicity for _ > 1=2 for degenerate noise. We also study the long time behaviour of the solutions tothe 2D stochastic quasi-geostrophic equation on T2 driven by real linear multiplicative noise and additive noise in the subcritical case by proving the existence of a random attractor.
Mon, 21 Jan 2013

14:15 - 15:15
Oxford-Man Institute

Contraction Rates for Bayesian Inverse Problems

SERGIOS AGAPIOU
(University of Warwick)
Abstract

Abstract: We consider the inverse problem of recovering u from a noisy, indirect observation We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0,together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available. We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0). The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A

Mon, 26 Nov 2012

15:45 - 16:45
Oxford-Man Institute

tbc

Karol Szczypkowski
Abstract
Mon, 26 Nov 2012

14:15 - 15:15
Oxford-Man Institute

Fractional Laplacian with gradient perturbations

Tomasz Jakubowski
Abstract

We consider the fractional Laplacian perturbed by the gradient operator b(x)\nabla for various classes of vector fields b. We construct end estimate the corresponding semigroup.

Mon, 19 Nov 2012

15:45 - 16:45
Oxford-Man Institute

Strong and weak solutions to stochastic Landau-Lifshitz equations

Zdzislaw Brzezniak
(University of York)
Abstract

I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.

Mon, 19 Nov 2012

14:15 - 15:15
Oxford-Man Institute

Google maps and improper Poisson line processes

WILFRID KENDALL
(University of Warwick)
Abstract

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.

Mon, 12 Nov 2012

15:45 - 16:45
Oxford-Man Institute

tbc

Wei Pan
(University of Oxford)
Abstract
Mon, 12 Nov 2012

14:15 - 15:15
Oxford-Man Institute

Towards a rigorous justification of kinetic theory: The gainless heterogeneous Boltzmann equation.

Florian Thiel
(University of Warwick)
Abstract

We study the asymptotic behavior of deterministic dynamics of many interacting particles with random initial data in the limit where the number of particles tends to infinity. A famous example is hard sphere flow, we restrict our attention to the simpler case where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density $f_0(u,v)$ depending on $d$-dimensional position $u$ and velocity $v$. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are $L^\infty$ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.