Oxford-Man Institute

Cubature is the business of describing a probability measure in terms of an empirical measure sharing its support with the original measure, of small support, and with identical integrals for a class of functions (eg polynomials with degree less than k). 

Applying cubature to already discrete sets of scenarios provides a powerful tool for scenario management and summarising data.  We refer to this process as recombination. It is a feasible operation in real time and has lead to high accuracy pde solvers.

The practical complexity of this operation has changed! By a factor corresponding to the dimension of the space of polynomials. 

We discuss the algorithm and give home computed examples of nested sparse grids with only positive weights in moderate dimensions (eg degree 1-8 in dimension 7).  Positive weights have significant advantage over signed ones when available.
 

Lawler, Schramm and Werner studied 2-point chordal restriction measures and gave several constructions using SLE tools.

It is possible to characterize 3-point chordal restriction measures in a similar manner. Their boundaries are SLE(8/3)-like curves with a slightly different drift term.

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   Stéphane Mallat

   Ecole Normale Superieure

Learning functionals in high dimension requires to find sources of regularity and invariants, to reduce dimensionality. Stability to actions of diffeomorphisms is a strong property satisfied by many physical functionals and most signal classification problems. We introduce a scattering operator in a path space, calculated with iterated multiscale wavelet transforms, which is invariant to rigid movements and stable to diffeomorphism actions. It provides a Euclidean embedding of geometric distances and a representation of stationary random processes. Applications will be shown for image classification and to learn quantum chemistry energy functionals.

In this talk we study the large time behaviour of some semilinear parabolic PDEs by a purely probabilistic approach. For that purpose, we show that the solution of a backward stochastic differential equation (BSDE) in finite horizon $T$ taken at initial time behaves like a linear term in $T$ shifted with a solution of the associated ergodic BSDE taken at inital time. Moreover we give an explicit rate of convergence: we show that the following term in the asymptotic expansion has an exponential decay. This is a Joint work with Ying Hu and Pierre-Yves Meyer from Rennes (IRMAR - France).

In this talk, we develop rough integration with jumps, offering a pathwise view on stochastic integration against cadlag processes.  A class of Marcus-like rough paths is introduced,which contains D. Williams’ construction of stochastic area for Lévy processes. We then established a Lévy–Khintchine type formula for the expected signature, based on“Marcus(canonical)"stochastic calculus. This calculus fails for non-Marcus-like Lévy rough paths and we treat the general case with Hunt’ theory of Lie group valued Lévy processes is made.

In this talk we will present some recent developments in the theory of ambit fields with a particular focuson limit theorems.
Ambit fields is a tempo-spatial class of models, which has been originally introduced by Barndorff-Nielsen and Schmiegel in the context of turbulence,
but found applications also in biology and finance. Its purely temporal analogue, Levy semi-stationary processes, has a continuous moving average structure
with an additional multiplicative random input (volatility or intermittency). We will briefly describe the main challenges of ambit stochastics, which
include questions from stochastic analysis, statistics and numerics. We will then focus on certain type of high frequency functionals typically called power variations.
We show some surprising non-standard limit theorems, which strongly depend on the driving Levy process. The talk is based on joint work with O.E. Barndorff-Nielsen, A. Basse-O'Connor,
J.M. Corcuera and R. Lachieze-Rey. 

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

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I will describe the Spatial Lambda-Fleming-Viot process, which is a model of evolution in a spatial continuum, and discuss the time and spatial scales on which selectively advantageous genes propagate through space. The appropriate scaling depends on the dimension of space, resulting in three distinct cases; d=1, d=2 and d>=3. In d=1 the limiting genealogy is the Brownian net whereas, by contrast, in d=2 local interactions give rise to a delicate damping mechanism and result in a finite limiting branching rate. This is joint work with Alison Etheridge and Daniel Straulino.

The theory of regularity structures allows one to give a meaning to several stochastic PDEs, including the Parabolic Anderson Model. So far, these equations have been considered on a torus. The goal of this talk is to explain how one can define the PAM on the whole space R^3. This is a joint work with Martin Hairer.