Past Stochastic Analysis Seminar

Abstract: It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales.

Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting. This is joint work with Thibault Jaisson.

: Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale.

This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault.

We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2.

We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility.

Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it.

This sheds light on why long memory of volatility has been widely accepted as a stylized fact.

Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.

This is joint work with Jim Gatheral and Thibault Jaisson.

We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms and prove an almost-Gaussian tail-estimate for the accumulated local p-variation functional, which has been introduced and studied by Cass, Litterer and Lyons. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao, and Chevyrev and Lyons.

The main goal of the talk is to set up a framework for constructing the likelihood for discretely observed RDEs. The main idea is to contract a function mapping the discretely observed data to the corresponding increments of the driving noise. Once this is known, the likelihood of the observations can be written as the likelihood of the increments of the corresponding noise times the Jacobian correction.

Constructing a function mapping data to noise is equivalent to solving the inverse problem of looking for the input given the output of the Ito map corresponding to the RDE. First, I simplify the problem by assuming that the driving noise is linear between observations. Then, I will introduce an iterative process and show that it converges in p-variation to the piecewise linear path X corresponding to the observations. Finally, I will show that the total error in the likelihood construction is bounded in p-variation.

Multiplicative chaos theory originated from the study of turbulence by Kolmogorov in the 1940s and it was mathematically founded by Kahane in the 1980s. Recently the theory has drawn much of attention due to its connection to SLEs and statistical physics.  In this talk I shall present some recent development of multiplicative chaos theory, as well as its applications to Liouville quantum gravity.

We consider the problem of large-scale regression where both the number of predictors, p, and the number of observations, n, may be in the order of millions or more. Computing a simple OLS or ridge regression estimator for such data, though potentially sensible from a purely statistical perspective (if n is large enough), can be a real computational challenge. One recent approach to tackling this problem in the common situation where the matrix of predictors is sparse, is to first compress the data by mapping it to an n by L matrix with L << p, using a scheme called b-bit min-wise hashing (Li and König, 2011). We study this technique from a theoretical perspective and obtain finite-sample bounds on the prediction error of regression following such data compression, showing how it exploits the sparsity of the data matrix to achieve good statistical performance. Surprisingly, we also find that a main effects model in the compressed data is able to approximate an interaction model in the original data. Fitting interactions requires no modification of the compression scheme, but only a higher-dimensional mapping with a larger L.
This is joint work with Nicolai Meinshausen (ETH Zürich).

The model of random interlacements is a one-parameter family of random subsets of $\Z^d$, which locally describes the trace of a simple random walk on a $d$-dimensional torus running up to time $u$ times its volume. Here, $u$ serves as an intensity parameter.

Its complement, the so-called vacant set, has been show to undergo a non-trivial percolation phase transition in $u$, i.e., there is $u_*(d)\in (0,\infty)$ such that for all $u<u_*(d)$ the vacant set has a unique infinite connected component (supercritical phase), while for $u>u_*(d)$ all connected components are finite.

So far all results regarding geometric properties of this infinite connected component have been proven under the assumption that $u$ is close to zero. 

I will discuss a recent result, which states that throughout most of the supercritical phase simple random walk on the infinite connected component is transient, provided that the dimension is high enough.

This is joint work with Alexander Drewitz

Abstract: L\'evy processes are increasingly popular for modelling stochastic process data with jump behaviour. In practice statisticians only observe discretely sampled increments of the process, leading to a statistical inverse problem. To understand the jump behaviour of the process one needs to make inference on the infinite-dimensional parameter given by the L\'evy measure. We discuss recent developments in the analysis of this problem, including in particular functional limit theorems for commonly used estimators of the generalised distribution function of the L\'evy measure, and their application to statistical uncertainty quantification methodology (confidence bands and tests). 

We consider the problem of minimising the commute or shuttle time for a diffusion between the endpoints of an interval. The control is the scale function for the diffusion. We show that the dynamic version of the problem has the same solution as the static version if we start at an end point and consider the much harder case where the starting point is in the interior.

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 We describe how cross-kernel matrices, that is, kernel matrices between the data and a custom chosen set of `feature spanning points' can be used for learning. The main potential of cross-kernel matrices is that (a) they provide Nyström-type speed-ups for kernel learning without relying on subsampling, thus avoiding potential problems with sampling degeneracy, while preserving the usual approximation guarantees and the attractive linear scaling of standard Nyström methods and (b) the use of non-square matrices for kernel learning provides a non-linear generalization of the singular value decomposition and singular features. We present a novel algorithm, Ideal PCA (IPCA), which is a cross-kernel matrix variant of PCA, showcasing both advantages: we demonstrate on real and synthetic data that IPCA allows to (a) obtain kernel PCA-like features faster and (b) to extract novel features of empirical advantage in non-linear manifold learning and classification.

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The Renormalization Group (RG) was pioneered by the physicist Kenneth Wilson in the early 70's and since then it has become a fundamental tool in physics. RG remains the most general philosophy for understanding how many models in statistical mechanics behave near their critical point but implementing RG analysis in a mathematically rigorous way remains quite challenging.

I will describe how analysis of RG flows translate into statements about continuum limits, universality, and cross-over phenomena - as a concrete example I will speak about some joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

Spectral volume and surface measures via the Dixmier trace for local symmetric Dirichlet spaces with Weyl type eigenvalue asymptotics

The purpose of this talk is to present the author's recent results of on an

operator theoretic way of looking atWeyl type Laplacian eigenvalue asymptotics

for local symmetric Dirichlet spaces.

For the Laplacian on a d-dimensional Riemannian manifoldM, Connes' trace

theorem implies that the linear functional  coincides with

(a constant multiple of) the integral with respect to the Riemannian volume

measure of M, which could be considered as an operator theoretic paraphrase

of Weyl's Laplacian eigenvalue asymptotics. Here  denotes a Dixmier trace,

which is a trace functional de_ned on a certain ideal of compact operators on

a Hilbert space and is meaningful e.g. for compact non-negative self-adjoint

operators whose n-th largest eigenvalue is comparable to 1/n.

The first main result of this talk is an extension of this fact in the framework

of a general regular symmetric Dirichlet space satisfying Weyl type asymptotics

for the trace of its associated heat semigroup, which was proved for Laplacians

on p.-c.f. self-simiar sets by Kigami and Lapidus in 2001 under a rather strong

assumption.

Moreover, as the second main result of this talk it is also shown that, given a

local regular symmetric Dirichlet space with a sub-Gaussian heat kernel upper

bound and a (sufficiently regular) closed subset S, a “spectral surface measure"

on S can be obtained through a similar linear functional involving the Lapla-

cian with Dirichlet boundary condition on S. In principle, corresponds to the

second order term for the eigenvalue asymptotics of this Dirichlet Laplacian, and

when the second order term is explicitly known it is possible to identify  For

example, in the case of the usual Laplacian on Rd and a Lipschitz hypersurface S,is a constant multiple of the usual surface measure on S.

Maximal couplings are couplings of Markov processes where the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian couplings are coupling strategies where neither process is allowed to look into the future of the other before making the next transition. These are easier to describe and play a fundamental role in many branches of probability and analysis. Hsu and Sturm proved that the reflection coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada proved that to have a MMC for Brownian motions on a Riemannian manifold, the manifold should have a reflection structure, and thus proved the first result connecting this purely probabilistic phenomenon (MMC) to the geometry of the underlying space.

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.

'We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated to the weak formulation of this problem is the limit of the value function associated to an unconstrained switching problem with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a constrained viscosity solution to a system of variational inequalities (SVI for short). We finally prove that the value function is the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.’

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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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.
 

   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.