Past Stochastic Analysis Seminar

21 November 2016
14:15
RICHARD SAMWORTH
Abstract

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.

 

  • Stochastic Analysis Seminar
14 November 2016
15:45
DAVID PROEMEL
Abstract

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.

 

  • Stochastic Analysis Seminar
14 November 2016
14:15
STEPHANE BOUCHERON
Abstract

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)

  • Stochastic Analysis Seminar
7 November 2016
15:45
MICHELE COGHI
Abstract

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.

  • Stochastic Analysis Seminar
7 November 2016
14:15
MARK GIROLAMI
Abstract

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. 

  • Stochastic Analysis Seminar
31 October 2016
15:45
BLANKA HORVATH
Abstract

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.

  • Stochastic Analysis Seminar
31 October 2016
14:15
MARTIN REDMANN
Abstract

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.

  • Stochastic Analysis Seminar
24 October 2016
15:45
Abstract

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.

  • Stochastic Analysis Seminar
24 October 2016
14:15
Abstract

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.

  • Stochastic Analysis Seminar
17 October 2016
15:45
YVAIN BRUNED
Abstract

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.

  • Stochastic Analysis Seminar

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