Stochastic Analysis Seminar
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Mon, 20/05 14:15 |
CAMILLE MALE (ENS Lyon) |
Stochastic Analysis Seminar  |
Oxford-Man Institute |
| Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence. A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries. Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices. The purpose of this talk is to give an introductory presentation of these notions. | |||
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Mon, 20/05 15:45 |
STEPHANE JAFFARD (universite PEC) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| Random wavelet series were introduced in the mid 90s as simple and flexible models that allow to take into account observed statistics of wavelet coefficients in signal and image processing. One of their most interesting properties is that they supply random processes whose pointwise regularity jumps form point to point in a very erratic way, thus supplying examples of multifractal processes. Interest in such models has been renewed recently under the spur of new applications coming from widely different fields; e.g. -in functional analysis, they allow to derive the regularity properties of “generic” functions in a given function space (in the sense of prevalence) -they offer toy examples on which one can check the accuracy of numerical algorithms that allow to derive the multifractal parameters associated with signals and images. We will give an overview of these properties, and we will focus on recent extensions whose sample paths are not locally bounded, and offer models for signals which share this property. | |||
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Mon, 03/06 14:15 |
PETER TANKOV (Universite Paris Diderot Paris 7) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| Jump processes, and Lévy processes in particular, are notoriously difficult to simulate. The task becomes even harder if the process is stopped when it crosses a certain boundary, which happens in applications to barrier option pricing or structural credit risk models. In this talk, I will present novel adaptive discretization schemes for the simulation of stopped Lévy processes, which are several orders of magnitude faster than the traditional approaches based on uniform discretization, and provide an explicit control of the bias. The schemes are based on sharp asymptotic estimates for the exit probability and work by recursively adding discretization dates in the parts of the trajectory which are close to the boundary, until a specified error tolerance is met. This is a joint work with Jose Figueroa-Lopez (Purdue). | |||
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Mon, 03/06 15:45 |
DOMINIQUE PICARD (Université Paris Diderot) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| Convergence of the Bayes posterior measure is considered in canonical statistical settings (like density estimation or nonparametric regression) where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions. A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived. | |||
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Mon, 10/06 14:15 |
PHILIPPE BRIAND (Universite Savoie) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| This talk is based on a joint work with Céline Labart. We are interested in this paper in the numerical simulation of solutions to Backward Stochastic Differential Equations. There are several existing methods to handle this problem and one of the main difficulty is always to compute conditional expectations. Even though our approach can also be applied in the case of the dynamic programmation equation, our starting point is the use of Picard's iterations that we write in a forward way In order to compute the conditional expectations, we use Wiener Chaos expansions of the underlying random variables. From a practical point of view, we keep only a finite number of terms in the expansions and we get explicit formulas. We will present numerical experiments and results on the error analysis. | |||
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Mon, 10/06 15:45 |
NI HAO (University of Oxford) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| In this talk, we consider the setting: a random realization of an evolving dynamical system, and explain how, using notions common in the theory of rough paths, such as the signature, and shuffle product, one can provide a new united approach to the fundamental problem of predicting the conditional distribution of the near future given the past. We will explain how the problem can be reduced to a linear regression and least squaresanalysis. The approach is clean and systematic and provides a clear gradation of finite dimensional approximations. The approach is also non-parametric and very general but still presents itself in computationally tractable and flexible restricted forms for concrete problems. Popular techniques in time series analysis such as GARCH can be seen to be restricted special cases of our approach but it is not clear they are always the best or most informative choices. Some numerical examples will be shown in order to compare our approach and standard time series models. | |||
