Past Stochastic Analysis Seminar

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function in a very directional way. Using an interpretation of this energy as a large deviation rate function for SLE_k as k goes to 0, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal i.e. of the same curve but viewed as going from B to A in D. In particular it measures how far does the chord differ from the hyperbolic geodesic. I will also discuss the relation between the energy of the curve with its regularity, some questions are still open. If time allows, I will present the Loewner energy for loops on the Riemann sphere, and open questions related to it as well.


 

The class of log-concave densities on $\mathbb{R}^d$ is a very natural infinite-dimensional generalisation of the class of Gaussian densities.  I will show that it also allows the statistician to have the best of both the parametric and nonparametric worlds, in that one can obtain a fully automatic density estimator in the class (via maximum likelihood), with no tuning parameters to choose.  I'll discuss its computation, methodological consequences and theoretical properties, and in particular very recent results on minimax rates of convergence and adaptation.

One of the central results in rough path theory is the local Lipschitz continuity of the solution map of a controlled differential equation called Ito-Lyons map. This continuity statement was obtained by T. Lyons in a q-variation resp. 1/q-Hölder type (rough path) metrics for any regularity 1/q>0. We extend this to a new class of Besov-Nikolskii type metrics with arbitrary regularity 1/q and integrability p, which particularly covers the aforementioned results as special cases. This talk is based on a joint work with Peter K. Friz.

 

This paper presents an adaptive version of the Hill estimator based on Lespki’s model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand’s concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of Extreme Value Theory: the quantile transform, Karamata’s representation of slowly varying functions, and Rényi’s characterisation for the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.

http://projecteuclid.org/euclid.ejs/1450456321  (joint work with Maud Thomas)

A system of interacting particles described by stochastic differential equations is considered. As opposed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (no interacting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of in viscid type, as opposed to the case when independent noises drive the different particles. Moreover, we use this result to derive a mean field approximation of the stochastic Euler equations for the vorticity of an incompressible fluid.

Ambitious mathematical models of highly complex natural phenomena are challenging to analyse, and more and more computationally expensive to evaluate. This is a particularly acute problem for many tasks of interest and numerical methods will tend to be slow, due to the complexity of the models, and potentially lead to sub-optimal solutions with high levels of uncertainty which needs to be accounted for and subsequently propagated in the statistical reasoning process. This talk will introduce our contributions to an emerging area of research defining a nexus of applied mathematics, statistical science and computer science, called "probabilistic numerics". The aim is to consider numerical problems from a statistical viewpoint, and as such provide numerical methods for which numerical error can be quantified and controlled in a probabilistic manner. This philosophy will be illustrated on problems ranging from predictive policing via crime modelling to computer vision, where probabilistic numerical methods provide a rich and essential quantification of the uncertainty associated with such models and their computation. 

We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention.
We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.

SPDEs with Lévy noise can be used to model chemical, physical or biological phenomena which contain uncertainties. When discretising these SPDEs in order to solve them numerically the problem might be of large order. The goal is to save computational time by replacing large scale systems by systems of low order capturing the main information of the full model. In this talk, we therefore discuss balancing related MOR techniques. We summarise already existing results and discuss recent achievements.

It is well-known that the stochastic heat equation on R^n has a Hölder continuous function-valued solution in the case n=1, and that in dimensions 2 and above the solution is not function-valued but is forced to take values in some wider space of distributions. So what happens if the space has, in some sense, a dimension in between 1 and 2? We turn to the theory of fractals in order to answer this question. It has been shown (Kigami, 2001) that there exists a class of self-similar sets on which natural Laplacians can be defined, and so an analogue to the stochastic heat equation can be posed. In this talk we cover the following questions: Is the solution to this equation function-valued? If so, is it Hölder continuous? To answer the latter we must first prove an analogue of Kolmogorov's celebrated continuity theorem for the self-similar sets that we are working on. Joint work with Ben Hambly.

We give an explicit scheme to reconstruct any C^1 curve from its signature. It is implementable and comes with detailed stability properties. The key of the inversion scheme is the use of a symmetrisation procedure that separates the behaviour of the path at small and large scales. Joint work with Terry Lyons.

The Regularity Structures introduced by Martin Hairer allow us to describe the solution of a singular SPDEs by a Taylor expansion with new monomials.  We present the two Hopf Algebras used in this theory for defining the structure group and the renormalisation group. We will point out the importance of recursive formulae with twisted antipodes.

Signature of a path provides a top down summary of the path as a driving signal. There have been substantial recent progress in reconstructing paths from its signature, (Lyons-Xu 2016, Geng 2016). In this talk, we focus on obtaining certain quantitative features of paths from their signatures. Hambly-Lyons' showed that the normalized limit of signature gives the length of a C^3 path. The result was recently extended by Lyons-Xu to C^1 paths. The extension of this result to bounded variation paths remains open. We will discuss this open problem.

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions and whose diffusivity has a sub-Riemannian structure. In the case where the endpoints agree, we discuss the convergence of the suitably rescaled fluctuations to a limiting diffusion loop, which is equal in law to the loop we obtain by taking the limiting process of the unconditioned rescaled diffusion processes and condition it to return to its starting point. The generator of the unconditioned limiting rescaled diffusion process can be described in terms of the original generator.

Malliavin calculus provides a framework to differentiate functionals defined on a Gaussian probability space with respect to the underlying noise. This allows to develop analysis on path space with infinite-dimensional generalisations of Fourier analysis, Sobolev spaces, etc from R^d. In this talk, we attempt to build a Lipschitz à la E. M. Stein (as opposed to Sobolev) function theory on rough path space. This framework allows to pathwise differentiate functionals on rough paths with respect to the underlying rough path. Time permitting, we show how to obtain Feynman-Kac-type representations for solutions to some high-order (>2) linear parabolic equations on R^d.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.

This talk will be about  Lévy processes on compact groups - discrete or continuous - and  two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by  reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments.     There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension.  I shall first sketch this correspondance for  Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior  of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops.  We characterize among them the master field defined by Thierry Lévy by a continuity property.   This is  a joint work with Guillaume Cébron and Franck Gabriel.

Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.

The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients

We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.

The planar Ising model is one of the simplest and most studied models in Statistical Mechanics. On one hand, it has a rich and interesting phase transition behaviour. On the other hand, it is "solvable" enough to allow for many rigorous and exact results. This, in particular, makes it one of the prime examples in Conformal Field Theory (CFT). In this talk, I will review my joint work with C. Hongler and D. Chelkak on the scaling limits of correlations in the planar Ising model at criticality. We prove that these limits exist, are conformally covariant and given by explicit formulae consistent with the CFT predictions. This may be viewed as a step towards a rigorous understanding of CFT in the case of the Ising model.TBC

We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model:It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle.These invariant measures are often called steady states.

The Einstein relation follows, at least for dimensions three and larger, from an expansion of the steady states as a function of the bias.

The talk is gase on joint work with Jan Nagel and Xiaoqin Guo

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.

The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$  In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.

Using Vovk's game-theoretic approach to mathematical finance and probability, it is possible to obtain new results in both areas.We first prove that one can make an arbitrarily large profit by investing in those one-dimensional paths which do not possess a local time of finite p-variation.  Additionally, we provide pathwise Tanaka formulas suitable for our local times and for absolutely continuous functions with sufficient regular derivatives. In the second part we derive a model-independent super-replication theorem in continuous time. Our result covers a broad range of exotic derivatives, including look-back options, discretely monitored Asian options, and options on realized variance.
 This talk is based on joint works with M. Beiglböck, A.M.G. Cox, M. Huesmann and N. Perkowski.