Past Stochastic Analysis Seminar

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.

The splitting-up method is a powerful tool to solve (SP)DEs by dividing the equation into a set of simpler equations that are easier to handle. I will speak about how such splitting schemes can be derived and extended by insights from the theory of rough paths.

Finally, I will discuss numerics for real-world applications that appear in the management of risk and engineering applications like nonlinear filtering.

Despite the ability of the stochastic volatility models along with their multivariate and multi-factor extension to describe the dynamics of the asset returns, these
models are very difficult to calibrate to market information. The recent financial crises, however, highlight that we can not use simplified models to describe the fincancial returns. Therefore, our statistical methodologies have to be improved. We propose a non parametricmethodology based on the use of the Fourier transform and the high frequency data which allows to estimate the diffusion and the leverage components of a general stochastic volatility model driven by continuous Brownian semimartingales. Our estimation procedure is based only on a pre-estimation of the Fourier coefficients of the volatility process and on the use of the Bohr convolution product as in Malliavin and Mancino 2009. This approach constitutes a novelty in comparison with the non-parametric methodologies proposed in the literature generally based on a pre-estimation of the spot volatility and in virtue of its definition it can be directly applied in the case of irregular tradingobservations of the price path an microstructure noise contaminations.

Sampling a $d$-dimensional continuous signal (say a semimartingale) $X:[0,T] \rightarrow \mathbb{R}^d$ at times $D=(t_i)$, we follow the recent papers [Gyurko-Lyons-Kontkowski-Field-2013] and [Lyons-Ni-Levin-2013] in constructing a lead-lag path; to be precise, a piecewise-linear, axis-directed process $X^D: [0,1] \rightarrow
\mathbb{R}^{2d}$ comprised of a past and future component. Lifting $X^D$ to its natural rough path enhancement, we can consider the question of convergence as
the latency of our sampling becomes finer.

The'signature', from the theory of differential equations driven by rough paths,
provides a very efficient way of characterizing curves. From a machine learning
perspective, the elements of the signature can be used as a set of features for
consumption by a classification algorithm.

Using datasets of letters, digits, Indian characters and Chinese characters, we
see that this improves the accuracy of online character recognition---that is
the task of reading characters represented as a collection of pen strokes.

We shall discuss the statistics of the eigenvalues of large random Hermitian matrices when the temperature is very high. In particular we shall focus on the transition from Wigner/Airy to Poisson regime.

Abstract: Given a sequence of random variables X_n that converge toward a Gaussian distribution, by looking at the next terms in the asymptotic E[exp(zX_n)] = exp(z^2 / 2) (1+ ...), one can often state a principle of moderate deviations. This happens in particular for sums of dependent random variables, and in this setting, it becomes useful to develop techniques that allow to compute the precise asymptotics of exponential generating series. Thus, we shall present a method of cumulants, which gives new results for the deviations of certain observables in statistical mechanics:

- the number of triangles in a random Erdos-Renyi graph;

- and the magnetization of the one-dimensional Ising model.

Market events such as order placement and order cancellation are examples of the complex and substantial flow of data that surrounds a modern financial engineer. New mathematical techniques, developed to describe the interactions of complex oscillatory systems (known as the theory of rough paths) provides new tools for analysing and describing these data streams and extracting the vital information. In this paper we illustrate how a very small number of coefficients obtained from the signature of financial data can be sufficient to classify this data for subtle underlying features and make useful predictions.

This paper presents financial examples in which we learn from data and then proceed to classify fresh streams. The classification is based on features of streams that are specified through the coordinates of the signature of the path. At a mathematical level the signature is a faithful transform of a multidimensional time series. (Ben Hambly and Terry Lyons \cite{uniqueSig}), Hao Ni and Terry Lyons \cite{NiLyons} introduced the possibility of its use to understand financial data and pointed to the potential this approach has for machine learning and prediction.

We evaluate and refine these theoretical suggestions against practical examples of interest and present a few motivating experiments which demonstrate information the signature can easily capture in a non-parametric way avoiding traditional statistical modelling of the data. In the first experiment we identify atypical market behaviour across standard 30-minute time buckets sampled from the WTI crude oil future market (NYMEX). The second and third experiments aim to characterise the market "impact" of and distinguish between parent orders generated by two different trade execution algorithms on the FTSE 100 Index futures market listed on NYSE Liffe.

Abstract:In this talk we consider a continuous time random walk $X$ on $\mathbb{Z}^d$ in an environment of random conductances taking values in $[0, \infty)$. Assuming that the law of the conductances is ergodic with respect to space shifts, we present a quenched invariance principle for $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme. Under the same conditions we also present a local limit theorem. For the proof some Hölder regularity of the transition density is needed, which follows from a parabolic Harnack inequality. This is joint work with J.-D. Deuschel and M. Slowik.