Past Stochastic Analysis Seminar

In this talk we will present some recent developments in the theory of ambit fields with a particular focuson limit theorems.
Ambit fields is a tempo-spatial class of models, which has been originally introduced by Barndorff-Nielsen and Schmiegel in the context of turbulence,
but found applications also in biology and finance. Its purely temporal analogue, Levy semi-stationary processes, has a continuous moving average structure
with an additional multiplicative random input (volatility or intermittency). We will briefly describe the main challenges of ambit stochastics, which
include questions from stochastic analysis, statistics and numerics. We will then focus on certain type of high frequency functionals typically called power variations.
We show some surprising non-standard limit theorems, which strongly depend on the driving Levy process. The talk is based on joint work with O.E. Barndorff-Nielsen, A. Basse-O'Connor,
J.M. Corcuera and R. Lachieze-Rey. 

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.

@email 

I will describe the Spatial Lambda-Fleming-Viot process, which is a model of evolution in a spatial continuum, and discuss the time and spatial scales on which selectively advantageous genes propagate through space. The appropriate scaling depends on the dimension of space, resulting in three distinct cases; d=1, d=2 and d>=3. In d=1 the limiting genealogy is the Brownian net whereas, by contrast, in d=2 local interactions give rise to a delicate damping mechanism and result in a finite limiting branching rate. This is joint work with Alison Etheridge and Daniel Straulino.

The theory of regularity structures allows one to give a meaning to several stochastic PDEs, including the Parabolic Anderson Model. So far, these equations have been considered on a torus. The goal of this talk is to explain how one can define the PAM on the whole space R^3. This is a joint work with Martin Hairer.

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.

In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges.

Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.

Based on joint work with Oliver Riordan.

We give a necessary and sufficient condition for a map defined on a compact, quasiconvex and simply-connected space to factor through a tree. This condition can be checked using currents. In particular if the target is some Euclidean space and the map is H\"older continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number. Moreover, this shows that if the target is the Heisenberg group equipped with the Carnot-Carath\'eodory metric and the H\"older exponent of the map is bigger than 2/3, the map factors through a tree.

The idea of A-free group, where A is a discrete ordered abelian group, has been introduced by Myasnikov, Remeslennikov and Serbin. It generalises the construction of free groups. A proof will be outlined that a group is A-free for some A if and only if it acts freely and without inversions on a \lambda-tree, where \lambda is an arbitrary ordered abelian group.

The Taylor expansion of a controlled differential equation suggests that the solution at time 1 depends on the driving path only through the latter's iterated integrals up to time 1, if the vector field is infinitely differentiable. Hambly and Lyons proved that this remains true for Lipschitz vector fields if the driving path has bounded total variation. We extend the Hambly-Lyons result for weakly geometric rough paths in finite dimension. Joint work with X. Geng, T. Lyons and D. Yang.    

We consider the numerical approximation of Kolmogorov equations arising in the context of option pricing under L\'evy models and beyond in a multivariate setting. The existence and uniqueness of variational solutions of the partial integro-differential equations (PIDEs) is established in Sobolev spaces of fractional or variable order.

Most discretization methods for the considered multivariate models suffer from the curse of dimension which impedes an efficient solution of the arising systems. We tackle this problem by the use of sparse discretization methods such as classical sparse grids or tensor train techniques. Numerical examples in multiple space dimensions confirm the efficiency of the described methods.

This work is joint with Peter Sarnak.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

We study infinite (random) systems of interacting particles living in a Euclidean space X and possessing internal parameter (spin) in R¹. Such systems are described by Gibbs measures on the space Γ(X,R¹) of marked configurations in X (with marks in R¹). For a class of pair interactions, we show the occurrence of phase transition, i.e. non-uniqueness of the corresponding Gibbs measure, in both 'quenched' and 'annealed' counterparts of the model.

A model of a financial market is complete if any payoff can be obtained as the terminal value of a self-financing trading strategy. It is well known that numerous models, for example stochastic volatility models, are however incomplete. We present conditions, which, in a general diffusion framework, guarantee that in such cases the market of primitive assets enlarged with an appropriate number of traded derivative contracts is complete. From a purely mathematical point of view we prove an integral representation theorem which guarantees that every local Q-martingale can be represented as a stochastic integral with respect to the vector of primitive assets and derivative contracts.

Abstract: We introduce a new framework for defining integration against rough path. This framework generalizes rough integral, and gives a natural explanation of some of the regularity requirements in rough path theory.

When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the sort time behaviour of $E(s)$. Furthermore, when the Minkowski dimension exists, finer geometric fluctuations can be recovered and $E(s)$ is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information.

When $\partial D$ is statistically self-similar, the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, $E(s)$ has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for small $s$, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale. In some cases, we can show that the fluctuations around this almost sure behaviour are governed by a central limit theorem, and conjecture that this is true more generally.

This is based on joint work with David Croydon and Ben Hambly.

We consider three nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a homogeneity test determines whether to accept or reject p=q; (2) Given a joint distribution p_xy over random variables x and y, an independence test investigates whether p_xy = p_x p_y, (3) Given a joint distribution over several variables, we may test for whether there exist a factorization (e.g., P_xyz = P_xyP_z, or for the case of total independence, P_xyz=P_xP_yP_z).

We present nonparametric tests for the three cases above, based on distances between embeddings of probability measures to reproducing kernel Hilbert spaces (RKHS), which constitute the test statistics (eg for independence, the distance is between the embedding of the joint, and that of the product of the marginals). The tests benefit from years of machine research on kernels for various domains, and thus apply to distributions on high dimensional vectors, images, strings, graphs, groups, and semigroups, among others. The energy distance and distance covariance statistics are also shown to fall within the RKHS family, when semimetrics of negative type are used. The final test (3) is of particular interest, as it may be used in detecting cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence, even when these variables have high dimension.

We will introduce a particle system interacting through a mean-field term that models the behavior of a network of excitatory neurons. The novel feature of the system is that the it features a threshold dynamic: when a single particle reaches a threshold, it is reset while all the others receive an instantaneous kick. We show that in the limit when the size of the system becomes infinite, the resulting non-standard equation of McKean Vlasov type has a solution that may exhibit a blow-up phenomenon depending on the strength of the interaction, whereby a single particle reaching the threshold may cause a macroscopic cascade. We moreover show that the particle system does indeed exhibit propagation of chaos, and propose a new way to give sense to a solution after a blow-up.

This is based on joint research with F. Delarue (Nice), E. Tanré (INRIA) and S. Rubenthaler (Nice).

Abstract: The signature of a path characterizes the non-commutative evolvements along the path trajectory. Nevertheless, one can extract local commutativities from the signature, thus leading to an inversion scheme.

It is well known that several solutions to the Skorokhod problem

optimize certain ``cost''- or ``payoff''-functionals. We use the

theory of Monge-Kantorovich transport to study the corresponding

optimization problem. We formulate a dual problem and establish

duality based on the duality theory of optimal transport. Notably

the primal as well as the dual problem have a natural interpretation

in terms of model-independent no arbitrage theory.

In optimal transport the notion of c-monotonicity is used to

characterize the geometry of optimal transport plans. We derive a

similar optimality principle that provides a geometric

characterization of optimal stopping times. We then use this

principle to derive several known solutions to the Skorokhod

embedding problem and also new ones.

This is joint work with Mathias Beiglböck and Alex Cox.

In this presentation, we focus on the decay rate of the expected signature of a stopped Brownian motion; more specifically we consider two types of the stopping time: the first one is the Brownian motion up to the first exit time from a bounded domain $\Gamma$, denoted by $\tau_{\Gamma}$, and the other one is the Brownian motion up to $min(t, \tau_{\Gamma\})$. For the first case, we use the Sobolev theorem to show that its expected signature is geometrically bounded while for the second case we use the result in paper (Integrability and tail estimates for Gaussian rough differential equation by Thomas Cass, Christian Litterer and Terry Lyons) to show that each term of the expected signature has the decay rate like 1/ \sqrt((n/p)!) where p>2. The result for the second case can imply that its expected signature determines the law of the signature according to the paper (Unitary representations of geometric rough paths by Ilya Chevyrev)

Simple probabilistic models for sequential data (text, music...), e.g., hidden Markov models, cannot capture some structures such as
long-term dependencies or combinations of simultaneous patterns and probabilistic rules hidden in the data. On the other hand, models such as
recurrent neural networks can in principle handle any structure but are notoriously hard to learn given training data. By analyzing the structure of
neural networks from the viewpoint of Riemannian geometry and information theory, we build better learning algorithms, which perform well on difficult
toy examples at a small computational cost, and provide added robustness.

Abstract: Polynomial preserving processes are defined as time-homogeneous Markov jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. These processes are attractive for financial modeling because of their tractability and robustness. In this work we study approximations of polynomial preserving processes with finite-state Markov processes via a moment-matching methodology. This approximation aims to exploit the defining property of polynomial preserving processes in order to reduce the complexity of the implementation of such models. More precisely, we study sufficient conditions for the existence of finite-state Markov processes that match the moments of a given polynomial preserving process. We first construct discrete time finite-state Markov processes that match moments of arbitrary order. This discrete time construction relies on the existence of long-run moments for the polynomial process and cubature methods over these moments. In the second part we give a characterization theorem for the existence of a continuous time finite-state Markov process that matches the moments of a given polynomial preserving process. This theorem illustrates the complexity of the problem in continuous time by combining algebraic and geometric considerations. We show the impossibility of constructing in general such a process for polynomial preserving diffusions, for high order moments and for sufficiently many points in the state space. We provide however a positive result by showing that the construction is possible when one considers finite-state Markov chains on lifted versions of the state space. This is joint work with Damir Filipovic and Martin Larsson.

I present some recent work with Bruce Driver and Christian Litterer on rough paths 'constrained’ to lie in a d - dimensional submanifold of a Euclidean space E. We will present a natural definition for this class of rough paths and then describe the (second) order geometric calculus which arises out of this definition. The talk will conclude with more advanced applications, including a rough version of Cartan’s development map.