Past Stochastic Analysis Seminar

Abstract: First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order asymptotic for renewal sequences with infinite mean and their consequences for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

Abstract: We consider different notions of rough paths on manifolds and study some of the relations between these definitions. Furthermore, we explore extensions to manifolds modelled along infinite dimensional Banach spaces.

Abstract: We consider random walks and Lévy processes in the free nilpotent Lie group as rough paths. For any p > 1, we completely characterise (almost) all Lévy processes whose sample paths have finite p-variation, provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process treated as a rough path, and give sufficient conditions under which a sequence of random walks converges weakly to a Lévy process in rough path topologies. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes. We demonstrate applications of our results to weak convergence of stochastic flows.

Abstract: A diffusion process on a Riemannian manifold whose generator is one half of the Laplacian is called a Brownian motion. The mean local time of Brownian motion on a hypersurface will be considered, as will the situation in which a Brownian motion is conditioned to arrive in a fixed submanifold at a fixed positive time. Doing so provides motivation for the remainder of the talk, in which a probabilistic formula for the integral of the heat kernel over a submanifold is proved and used to deduce lower bounds, an asymptotic relation and derivative estimates applicable to the conditioned process.

Abstract: The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle. This is a joint work with Gaoyue Guo and Nizar Touzi.

Abstract: Joint work with Oleg Zaboronsky (Warwick).

Some one dimensional nearest neighbour particle systems are examples of Pfaffian point processes - where all intensities are determined by a single kernel.In some cases these kernels have appeared in the random matrix literature (where the points are the positions of eigenvalues). We are attempting to use random matrix tools on the particle sytems, and particle tools on the random matrices.

Abstract:   Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points), when properly centered and scaled, converges to the Tracy-Widom distribution. Later Johansson showed that all maximal paths lie within the strip of width $n^{\frac{2}{3} +\epsilon}$ around the diagonal with probability tending to 1 as $n \to \infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma< \frac{2}{3}$.

 

TBC

There is a simple way to “glue together” a coupled pair of continuum random trees to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the “interface” between the trees). We present an explicit and canonical way to embed the sphere into the Riemann sphere. In this embedding, the measure is Liouville quantum gravity with parameter gamma in (0,2), and the curve is space-filling version of SLE with kappa=16/gamma^2. Based on joint work with Bertrand Duplantier and Scott Sheffield

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

Abstract:In the first part of the talk, using classical hypergeometric identities, I will compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. In the second part of the talk, I will consider the two-dimensional Markov process based on the stable Lévy process and its area process. I will give two explicit formulae for the harmonic measure of the split complex plane. These formulae allow to compute the persistence exponent of the stable area process, solving a problem raised by Zhan Shi. This is based on two joint works with Christophe Profeta.

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.

tba

tba

 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.

tba