Statistics for Stochastic Processes: SDEs, SPDEs and concentration of measure

Abstract: A classical result due to Aida and Stroock asserts that the log-Sobolev inequality implies subgaussian moment estimates for sufficiently regular functions, generalizing L_p-Poincare inequality for the Gaussian measure, due to Maurey and Pisier. In many interesting examples the log-Sobolev inequality does not hold or holds with a rather bad constant and in order to derive concentration results for Lipschitz functions one replaces it with a suitably modified weaker version. I will discuss equivalence of this modified inequality with a suitable family of Beckner-type inequalities and show how an adaptation of an argument introduced by Boucheron-Bousquet-Lugosi-Massart in a context of independent random variables, allows to obtain subgaussian moment estimates also in this less restrictive setting, which leads to concentration results for non-Lipschitz functions. If time permits, I will discuss examples with potential statistical applications, including the Poisson space, the Ising model and certain measures on sections of the discrete cube. Based on joint work with Bartlomiej Polaczyk and Michal Strzelecki.

Abstract: The problem of parameter estimation in a general second order linear stochastic partial differential equation (SPDE) is considered. One trajectory of the solution to the SPDE is observed continuously in time and averaged in space over a small window at multiple locations. Estimators for the diffusivity, transport and reaction coefficients are constructed. These estimators are shown to be minimax rate optimal by proving an explicit lower bound in the asymptotic regime where the spatial window shrinks to zero and with a growing number of observations. Interestingly, the rate of convergence depends on the differential order in which the respective coefficient appears, with the fastest rate achieved for the diffusivity coefficient and the slowest rate for the reaction terms. The proof for the lower bounds relies on an explicit analysis of the reproducing kernel Hilbert spaces for the observed Gaussian processes, and may be of independent interest.

Abstract: In this talk, we study the problem of estimating the parameters of a coupled MvKean-Vlasov SDE based on observations of the corresponding particle system. We derive the convergence rates for an estimator of the potential in a semi-parametric setting and discuss their optimality.

Abstract: Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, it was proved by Cheridito that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on a second surprise in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

This is based on joint work with Thomas Delerue (TU Munich) and Fabian Mies (RWTH Aachen).

Abstract: Motivated by problems from statistical analysis for discretely sampled SPDEs, we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. We prove a new central limit theorem for some power variations of the iterated integrals of a fractional Brownian motion (fBm) and consequently apply them to estimation of the drift and volatility coefficients of semilinear stochastic partial differential equations driven by an additive Gaussian noise white in time and possibly colored in space. In particular, we show that approximating naively derivatives by finite differences in certain estimators may introduce a nontrivial bias that we compute explicitly.

Abstract: We propose a novel asymptotic regime for statistical inference for SPDEs, in which the diffusivity of the system tends to zero. The maximum likelihood estimator for the reaction term in semi-linear SPDEs is consistent, satisfies a central limit theorem and is asymptotically efficient. In contrast to most of the existing literature, we do not require that the time horizon increases to achieve consistency. A higher order of the leading differential operator results in a faster convergence rate. The fluctuations of the empirical Fisher information around its mean are controlled using the infinite-dimensional Poincaré inequality. Both parametric and non-parametric estimation is discussed. Numerical examples illustrate the key findings.

[1] Gaudlitz, S., Reiß, M. (Working paper) Estimation for the reaction term in semi-linear SPDEs under small diffusivity

Abstract: In this paper, we consider the problem of joint parameter estimation for drift and volatility coefficients of a stochastic McKean-Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution of the process and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval [0, T], we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step ($\Delta_n$) goes to 0 and the number of particles N goes to $\infty$, and asymptotically normal when additionally the condition $\Delta_n N \rightarrow 0$ holds. The talk is based on joint work with C. Amorino, V. Pilipauskaite and M. Podolskij.

Abstract: We revisit an old problem: estimate non-parametrically the drift and diffusion coefficient of a diffusion process from discrete data $(X_0,X_D, X_{2D}, \ldots, X_{ND})$. The novelty are: (i) we work in a multivariate setting: only few results have been obtained in this case from discrete data (and, to the best of our knowledge, no results for the diffusion coefficient) (ii) the sampling scheme is high frequency but arbitrarily slow: $D=D_N \rightarrow 0$ and $N_D_N^q \rightarrow 0 from some possibly arbitrarily large $q$ (à la Kessler) and (iii) the process lies in a (not necessarily convex, not necessarily bounded) domain in $\mathbb R^d$ with reflection at the boundary. (In particular we recover the case of a bounded domain or the whole Euclidean space $R^d$.) We conduct a (relatively) standard minimax — adaptive — program for integrated squared error loss over bounded domains (and more losses in the simpler case of the drift) over standard smoothness classes plus some other miscellanies. When $ND^2 \rightarrow 0$ and in the special case of the conductivity equation over a bounded domain, we actually obtain contraction rates in squared error loss in a nonparametric Bayes setting. The main difficulty here lies in controlling small balll probabilities for the likelihood ratios; we develop small time expansions of the heat kernel with a bit of Riemannian geometry to control adequate perturbations in KL divergence, using old ideas of Azencott and others. That last part is joint with K. Ray and R. Nickl.

Although these questions could have been methodologically addressed almost two decades ago, we heavily rely on the substantial progress that have been made since in the domain to clarify and quantify the stability of ergodic averages via concentration chaining techniques and explicit mixing bounds + other features (Dirksen, Nickl, Paulin, Ray, Reiss, Strauch and many others).

Abstract: An algorithm performing short-term predictions of the flux of vehicles on a stretch of road is presented, using past measurements of the flux. This algorithm is based on a physics-aware recurrent neural network which combines the discretization of a macroscopic traffic flow model by the so-called Traffic Reaction Model with the prediction of the space-time dependent traffic parameters by a succession of LSTM and simple RNNs. Besides, the algorithm yields smoothing of its inputs which allows it to handle noisy and faulty raw data without preprocessing. The algorithm is tested on raw flux measurements obtained from loop detectors. The talk is based on joint work with Mike Pereira and Balázs Kulcsár.

Abstract: We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups. This is a joint work with Xavier Erny (Polytechnique) and Dasha Loukianova (Evry).

Abstract: Infinitely divisible processes (IDP) form a very large class of stochastic processes including Levy and additive processes, Ornstein-Uhlenbeck processes or moving average processes. In this talk I would like to describe a method of characterizing the suprema of such processes using so-called chaining- the method known mainly due to the famous Fernique-Talagrand's Majorising Measure Theorem. I will briefly recall the history of the chaining and try to explain the results on IDPs in this context. Also, how the same method applies to the characterization of the suprema of empirical processes. The talk is based on the joint paper with Witold Bednorz: "The suprema of infinitely divisible processes".

Abstract: TBA

Abstract: This talk is based on results obtained jointly with Petr Čoupek and Martin Ondreját, cf. [1].
The problem of integration of temporal functions taking values in Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is discussed. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter, the Rosenblatt process or, more generally, the Hermite processes. The class of Banach spaces satisfying our conditions includes the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. Domains of the Wiener integrals on both bounded and unbounded intervals is characterized for both scalar and cylindrical fractional processes. In general, the integrands take values in the space of gamma-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. An equivalent characterization in terms of a pointwise kernel of the integrand is also presented if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. These results are applied to stochastic convolution integrals. In particular, solutions of the heat equation with distributed noise of low time regularity and of the heat equation with boundary noise are discussed .

[1]. P. Čoupek, B. Maslowski and M. Ondreját, Stochastic integration with respect to fractional processes in Banach spaces, J. Funct. Anal. 282 (2022), no. 8, Paper No. 109393, 62 pp.

Abstract: In this talk we will review some recent results on Gaussian fluctuations of spatial averages of the one-dimensional stochastic heat equation driven by a space-time Gaussian white noise, that have been established by a combination of Stein’s method for normal approximations and Malliavin calculus techniques. We will also discuss the rate of the convergence to zero of the uniform distance between the density of spatial averages of the solution, properly normalized, and the standard normal density.

Abstract: We consider the problem of estimating the drift of an abstract linear stochastic parabolic evolution equation perturbed by small observation noise. Using a kernel smoothing approach, we construct a modified maximum likelihood estimator and study its asymptotic properties. More precisely, we find an optimal tradeoff between exploiting small-scale spatial information and smoothing out the observation noise.
This Talk is based on work in progress with Markus Reiß.

Abstract: We investigate the problem of estimating the drift parameter of a high-dimensional Lévy-driven Ornstein–Uhlenbeck process under sparsity constraints. It is shown that both Lasso and Slope estimators achieve the minimax optimal rate of convergence (up to numerical constants), for tuning parameters chosen independently of the confidence level. The results are non-asymptotic and hold both in probability and conditional expectation with respect to an event resembling the restricted eigenvalue condition.
Based on joint work with Niklas Dexheimer.

Abstract: We study a linear SPDE on a bounded domain driven a stochastic noise process which is white in time and possibly colored in space. We aim for bridging the gap between two popular observations schemes studied for statistics for SPDEs, namely, discrete observations and local measurements. To this end, we extend the local measurements approach to kernels of distribution type. In particular, the discrete Laplacian will allow for analyzing estimators based on discrete observations in arbitrary space dimensions.
Joint work with Randolf Altmeyer and Florian Hildebrandt

Abstract: We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the usefulness of the results by applying them in the context of high-dimensional drift estimation and Langevin MCMC for moderately heavy-tailed target densities.

Abstract: We consider the fractional stochastic heat equation driven by a nonlinear Gaussian space-time white noise and we analyze its mild solution. We actually study the limit behavior of the spatial quadratic variation of its mild solution both in the linear and nonlinear noise cases by obtaining the exact limit of this quadratic variation. We apply these results to parameter estimation. More precisely, we construct an estimator for the drift parameter of the fractional stochastic heat equation with nonlinear noise, which is defined in terms of the quadratic variation and it is based on the observation of the solution at a fixed time and at discrete points in space. The proofs are based on the relation between the solution to the linear fractional stochastic heat equation and the fractional Brownian motion and on a sharp analysis of the Green kernel associated to the fractional Laplacian operator.

Abstract: We consider parameter estimation for a linear parabolic second-order stochastic partial differential equation (SPDE) in two space dimensions with a small dispersion parameter based on high frequency spatio-temporal data with respect to (w.r.t.) time and space. A driving processes of the SPDE is a Q-Wiener process, see [4]. There are several studies on parameter estimation of a linear parabolic second-order SPDE in one space dimension driven by the cylindrical Wiener process based on high frequency spatio-temporal data observed on a fixed region, see [1] and [2]. Minimum contrast estimators for unknown parameters of the SPDE in one space dimension with a small noise are proposed by [3] and the asymptotic normality of the estimators is shown. In this talk, we first obtain minimum contrast estimators of three coefficient parameters of the SPDE based on the thinned data w.r.t. space points. Secondly, we approximate a coordinate process of the SPDE using the minimum contrast estimators. Note that the coordinate process is the Ornstein-Uhlenbeck process with a small noise. Finally, we propose parametric adaptive estimators of the rest of unknown parameters of the SPDE using the approximate coordinate process. This is a joint work with Yozo Tonaki (Osaka University) and Yusuke Kaino (Kobe University).

[1] Bibinger, M. and Trabs, M. (2020). Volatility estimation for stochastic PDEs using high- frequency observations, Stochastic Processes and their Applications, 130, 3005–3052.
[2] Hildebrandt, F. and Trabs, M. (2021). Parameter estimation for SPDEs based on discrete observations in time and space, Electronic Journal of Statistics, 15, 2716–2776.
[3] Kaino, Y. and Uchida, M. (2021). Adaptive estimator for a parabolic linear SPDE with a small noise, Japanese Journal of Statistics and Data Science, 4, 513–541.
[4] Tonaki, Y., Kaino Y. and Uchida, M. (2022). Parameter estimation for linear parabolic SPDEs in two space dimensions based on high frequency data, arXiv preprint arXiv:2201.09036.

Abstract: TBA