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We consider linear models with scalar responses and covariates from a separable Hilbert space. The aim is to detect change points in the error distribution, based on sequential residual empirical distribution functions. Expansions for those estimated functions are more challenging in models with infinite-dimensional covariates than in regression models with scalar or vector-valued covariates due to a slower rate of convergence of the parameter estimators. Yet the suggested change point test is asymptotically distribution-free and consistent for one-change point alternatives. In the latter case we also show consistency of a change point estimator.

In this paper, we study the stochastic calculus for the theta process. The theta process is a stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums. It has several properties in common with Brownian motion such as its H\"older regularity, uncorrelated increments and quadratic variation. However, crucially, we show that the theta process is not a semimartingale, making It\^o calculus techniques inapplicable. Instead, we use the celebrated rough paths theory to develop the stochastic calculus for the theta process. We do so by constructing the iterated integrals - the ``rough path" - above the theta process. Rough paths theory takes a signal and its iterated integrals and produces a vast and robust theory of stochastic differential equations. In addition, the rough path we construct can be described in terms of higher rank theta sums., Comment: 61 pages. Applications of rough paths theory to number theory

Decentralized stochastic control problems with local information involve problems where multiple agents and subsystems which are coupled via dynamics and/or cost are present. Typically, however, the dynamics of such couplings is complex and difficult to precisely model, leading to questions on robustness in control design. Additionally, when such a coupling can be modeled, the problem arrived at is typically challenging and non-convex, due to decentralization of information. In this paper, we develop a robustness framework for optimal decentralized control of interacting agents, where we show that a decentralized control problem with interacting agents can be robustly designed by considering a risk-sensitive version of non-interacting agents/particles. This leads to a tractable robust formulation where we give a bound on the value of the cost function in terms of the risk-sensitive cost function for the non-interacting case plus a term involving the ``strength" of the interaction as measured by relative entropy. We will build on Gaussian measure theory and an associated variational equality. A particular application includes mean-field models consisting of (a generally large number of) interacting agents which are often hard to solve for the case with small or moderate numbers of agents, leading to an interest in effective approximations and robustness. By adapting a risk-sensitivity parameter, we also robustly control a non-symmetrically interacting problem with mean-field cost by one which is symmetric with a risk-sensitive criterion, and in the limit of small interactions, show the stability of optimal solutions to perturbations.

The theory of stochastic representations of solutions to elliptic and parabolic PDE has been extensive. However, the theory for hyperbolic PDE is notably lacking. In this short note we give a stochastic representation for solutions of hyperbolic PDE., Comment: 5 pages. To appear in J. Stoch. Anal

In this article we show a robustness theorem for controlled stochastic differential equations driven by approximations of Brownian motion. Often, Brownian motion is used as an idealized model of a diffusion where approximations such as Wong-Zakai, Karhnen-Lo\`eve or fractional Brownian motion are often seen as more physical. However, there has been extensive literature on solving control problems driven by Brownian motion and little on control problems driven by more realistic models that are only approximately Brownian. The question of robustness naturally arises from such approximations. We show robustness using rough paths theory, which allows for a pathwise theory of stochastic differential equations. To this end, in particular, we show that within the class of Lipschitz continuous control policies, an optimal solution for the Brownian idealized model is near optimal for a true system driven by a non-Brownian (but near-Brownian) noise.

Rough paths theory allows for a pathwise theory of solutions to differential equations driven by highly irregular signals. The fundamental observation of rough paths theory is that if one can define "iterated integrals" above a signal, then one can construct solutions to differential equations driven by the signal. The typical examples of the signals of interest are stochastic processes such as (fractional) Brownian motion. However, rough paths theory is not inherently random and therefore can treat irregular deterministic driving signals such as a (multivariate) Weierstrass function. To the authors' best knowledge, no explicit construction of a rough path (the "iterated integrals") above a multivariate Weierstrass function has been constructed, nor has there been an explicit solution to a differential equation driven by a multivariate Weierstrass function. This note supplies a construction of a rough path above a multivariate Weierstrass function. We conclude with some illustrations and some examples of solving differential equations driven by rough Weierstrass functions., Comment: To appear in C. R. Math. Acad. Sci. Paris

In Selk and Gertheiss (2022) a nonparametric prediction method for models with multiple functional and categorical covariates is introduced. The dependent variable can be categorical (binary or multi-class) or continuous, thus both classification and regression problems are considered. In the paper at hand the asymptotic properties of this method are developed. A uniform rate of convergence for the regression / classification estimator is given. Further it is shown that, asymptotically, a data-driven least squares cross-validation method can automatically remove irrelevant, noise variables.

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the ``most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the ``most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and $\Gamma$-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations., Comment: To appear in ALEA Lat. Am. J. Probab. Math. Stat

We consider nonparametric prediction with multiple covariates, in particular categorical or functional predictors, or a mixture of both. The method proposed bases on an extension of the Nadaraya-Watson estimator where a kernel function is applied on a linear combination of distance measures each calculated on single covariates, with weights being estimated from the training data. The dependent variable can be categorical (binary or multi-class) or continuous, thus we consider both classification and regression problems. The methodology presented is illustrated and evaluated on artificial and real world data. Particularly it is observed that prediction accuracy can be increased, and irrelevant, noise variables can be identified/removed by `downgrading' the corresponding distance measures in a completely data-driven way., Comment: accepted for publication

In this article, we show that every centered Gaussian measure on an infinite dimensional separable Fr\'{e}chet space $X$ over $\mathbb R$ admits some full measure Banach intermediate space between $X$ and its Cameron-Martin space. We provide a way of generating such spaces and, by showing a partial converse, give a characterization of Banach intermediate spaces. Finally, we show an example of constructing an $\alpha$-H\"older intermediate space in the space of continuous functions, $\mathcal C_0[0, 1]$ with the classical Wiener measure., Comment: Comments welcome!

In this article, we prove a Feynman-Kac type result for a broad class of second order ordinary differential equations. The classical Feynman-Kac theorem says that the solution to a broad class of second order parabolic equations is the mean of a particular diffusion. In our situation, we show that the solution to a system of second order ordinary differential equations is the mode of a diffusion, defined through the Onsager-Machlup formalism. One potential utility of our result is to use Monte Carlo type methods to estimate the solutions of ordinary differential equations. We conclude with examples of our result illustrating its utility in numerically solving linear second order ODEs., Comment: Comments welcome!

In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such processes on intervals $[a,b]$ where $b-a$ is either less than the increment or twice the increment, assuming stationarity of the increments. As a chief example, we consider fractional Brownian motion and fractional Gaussian noise for $H\geq 1/2$., Comment: Comments Welcome!

This paper studies constrained information projections on Banach spaces with respect to a Gaussian reference measure. Specifically our interest lies in characterizing projections of the reference measure, with respect to the KL-divergence, onto sets of measures corresponding to changes in the mean (or {\it shift measures}). As our main result, we give a portmanteau theorem that characterizes the relationship among several different formulations of this problem. In the general setting of Gaussian measures on a Banach space, we show that this information projection problem is equivalent to minimization of a certain Onsager-Machlup (OM) function with respect to an associated stochastic process. We then construct several reformulations in the more specific setting of classical Wiener space. First, we show that KL-weighted optimization over shift measures can also be expressed in terms of an OM function for an associated stochastic process that we are able to characterize. Next, we show how to encode the feasible set of shift measures through an explicit functional constraint by constructing an appropriate penalty function. Finally, we express our information projection problem as a calculus of variations problem, which suggests a solution procedure via the Euler-Lagrange equation. We work out the details of these reformulations for several specific examples., Comment: To Appear in Applied Mathematics and Optimization

Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $\int y \, dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $\int f(X) dX$.

In this work, we consider a multivariate regression model with one-sided errors. We assume for the regression function to lie in a general H\"{o}lder class and estimate it via a nonparametric local polynomial approach that consists of minimization of the local integral of a polynomial approximation lying above the data points. While the consideration of multivariate covariates offers an undeniable opportunity from an application-oriented standpoint, it requires a new method of proof to replace the established ones for the univariate case. The main purpose of this paper is to show the uniform consistency and to provide the rates of convergence of the considered nonparametric estimator for both multivariate random covariates and multivariate deterministic design points. To demonstrate the performance of the estimators, the small sample behavior is investigated in a simulation study in dimension two and three.

In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time $\lfloor ns_0\rfloor$, $s_0\in(0,1)$, and our aim is to estimate the (rescaled) change point $s_0$. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of $O_P(n^{-1})$ which in this case is clearly optimal as there are only $n$ points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cram\'er-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.