Publications

Monographs

  • D. Belomestny, J. Schoenmakers, Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance, Macmillan Publishers Ltd., London, 2018, 364 pages, (Monograph Published), DOI 10.1057/978-1-137-03351-2 .

  • H.-G. Bartel, H.-J. Mucha, Chapter 2: Incomparability/Inequality Measures and Clustering, in: Partial Order Concepts in Applied Sciences, M. Fattore, R. Brüggemann, eds., Springer International Publishing, New York, 2017, pp. 21--34, (Chapter Published).

Articles in Refereed Journals

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic model for LFP-electrodes, Continuum Mechanics and Thermodynamics, published online on 05.02.2018, DOI 10.1007/s00161-018-0629-7 .
    Abstract
    In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

  • TH. Koprucki, M. Kohlhase, K. Tabelow, D. Müller, F. Rabe, Model pathway diagrams for the representation of mathematical models, Optical and Quantum Electronics, (2018), published online on 23.01.2018, DOI 10.1007/s11082-018-1321-7 .
    Abstract
    Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the van Roosbroeck system describing the carrier transport in semiconductors by drift and diffusion. We introduce an approach for the block-based composition of models from simpler components.

  • F. Anker, Ch. Bayer, M. Eigel, M. Ladkau, J. Neumann, J.G.M. Schoenmakers, SDE based regression for random PDEs, SIAM Journal on Scientific Computing, 39 (2017), pp. A1168--A1200.
    Abstract
    A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.

  • F. Anker, Ch. Bayer, M. Eigel, J. Neumann, J.G.M. Schoenmakers, A fully adaptive interpolated stochastic sampling method for linear random PDEs, International Journal for Uncertainty Quantification, 7 (2017), pp. 189--205, DOI 10.1615/Int.J.UncertaintyQuantification.2017019428 .
    Abstract
    A numerical method for the fully adaptive sampling and interpolation of PDE with random data is presented. It is based on the idea that the solution of the PDE with stochastic data can be represented as conditional expectation of a functional of a corresponding stochastic differential equation (SDE). The physical domain is decomposed subject to a non-uniform grid and a classical Euler scheme is employed to approximately solve the SDE at grid vertices. Interpolation with a conforming finite element basis is employed to reconstruct a global solution of the problem. An a posteriori error estimator is introduced which provides a measure of the different error contributions. This facilitates the formulation of an adaptive algorithm to control the overall error by either reducing the stochastic error by locally evaluating more samples, or the approximation error by locally refining the underlying mesh. Numerical examples illustrate the performance of the presented novel method.

  • M. Redmann, P. Benner, Singular perturbation approximation for linear systems with Lévy noise, Stochastics and Dynamics, (2017), published online on 10.08.2017, DOI 10.1142/S0219493718500338 .
    Abstract
    To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations, we consider the singular perturbation approximation as a model order reduction technique in this paper. This approach is well-known from deterministic control theory and here we generalize it for controlled linear systems with Lévy noise. Additionally, we discuss properties of the reduced order model, provide an error bound, and give some examples to demonstrate the quality of this model order reduction technique.

  • A. Andresen, V. Spokoiny, Convergence of an alternating maximization procedure, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 53 (2017), pp. 389--429, DOI 10.1214/15-AIHP720 .

  • A. Anikin, A. Gasnikov, P. Dvurechensky, A. Turin, A. Chernov, Dual approaches to the minimization of strongly convex functionals with a simple structure under affine constraints, Computational Mathematics and Mathematical Physics, 57 (2017), pp. 1262--1276.

  • D. Belomestny, R. Hildebrand, J.G.M. Schoenmakers, Optimal stopping via pathwise dual empirical maximisation, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, published online on 8.11.2017, DOI 10.1007/s00245-017-9454-9 .
    Abstract
    The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the path-wise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

  • D. Belomestny, H. Mai, J.G.M. Schoenmakers, Generalized Post--Widder inversion formula with application to statistics, Journal of Mathematical Analysis and Applications, 455 (2017), pp. 89--104.
    Abstract
    In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known Post-Widder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variance-mean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized Post-Widder formula, derive bounds for its root mean square error and give a brief numerical example.

  • S. Bürger, P. Mathé, Discretized Lavrent'ev regularization for the autoconvolution equation, Applicable Analysis. An International Journal, 96 (2017), pp. 1618--1637, DOI 10.1080/00036811.2016.1212336 .
    Abstract
    Lavrent?ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent?ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333?348. Here this study is extended by considering discretization of the Lavrent?ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

  • G. Cannizzaro, P. Friz, P. Gassiat, Malliavin calculus for regularity structures: The case of gPAM, Journal of Functional Analysis, 272 (2017), pp. 363--419, DOI 10.1016/j.jfa.2016.09.024 .
    Abstract
    Malliavin calculus is implemented in the context of Hairer (2014) [16]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust, and purely deterministic, translation operator, in -directions, between ?models?. In the concrete context of the generalized parabolic Anderson model in 2D ? one of the singular SPDEs discussed in the afore-mentioned article ? we establish existence of a density at positive times.

  • J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Stochastic Processes and their Applications, (2017), published online on 21.9.2017, DOI 10.1016/j.spa.2017.09.010 .
    Abstract
    We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean?Vlasov type limit, as shown in two corollaries.

  • A. Gasnikov, E. Gasnikova, P. Dvurechensky, A. Mohammed, E. Chernousova, About the power law of the PageRank vector distribution. Part 1. Numerical methods for finding the PageRank vector, Sibirskii Zhurnal Vychislitelnoi Matematiki. Siberian Journal of Numerical Mathematics. Rossiiskaya Akademiya Nauk. Sibirskoe Otdelenie, 20 (2017), pp. 359--376.

  • Y. Nesterov, V. Spokoiny, Random gradient-free minimization of convex functions, Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics, 17 (2017), pp. 527--566.
    Abstract
    Summary: In this paper, we prove new complexity bounds for methods of convex optimization based only on computation of the function value. The search directions of our schemes are normally distributed random Gaussian vectors. It appears that such methods usually need at most nn times more iterations than the standard gradient methods, where nn is the dimension of the space of variables. This conclusion is true for both nonsmooth and smooth problems. For the latter class, we present also an accelerated scheme with the expected rate of convergence O(n2k2)O(n2k2), where kk is the iteration counter. For stochastic optimization, we propose a zero-order scheme and justify its expected rate of convergence O(nk1/2)O(nk1/2). We give also some bounds for the rate of convergence of the random gradient-free methods to stationary points of nonconvex functions, for both smooth and nonsmooth cases. Our theoretical results are supported by preliminary computational experiments.

  • CH. Bayer, M. Siebenmorgen, R. Tempone, Smoothing the payoff for efficient computation of basket option prices, Quantitative Finance, (2017), published online on 20.7.2017, DOI 10.1080/14697688.2017.1308003 .
    Abstract
    We consider the problem of pricing basket options in a multivariate Black Sc- holes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non- smooth integrands. Due to this lack of regularity, higher order numerical integration tech- niques may not be directly available, requiring the use of methods like Monte Carlo specif- ically designed to work for non-regular problems. We propose to use the inherent smooth- ing property of the density of the underlying in the above models to mollify the payo ff function by means of an exact conditional expectation. The resulting conditional expec- tation is unbiased and yields a smooth integrand, which is amenable to the e ffi cient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 25.

  • CH. Bayer, U. Horst, J. Qiu, A functional limit theorem for limit order books with state dependent price dynamics, The Annals of Applied Probability, 27 (2017), pp. 2753-2806.
    Abstract
    We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity, the impact of an individual order arrival on the book as well as the tick size tend to zero. The limits of the standing buy and sell volume densities are described by two linear stochastic partial differential equations, which are coupled with a two-dimensional reflected Brownian motion that is the limit of the best bid and ask price processes.

  • P. Friz, J. Diehl, P. Gassiat, Stochastic control with rough paths, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 75 (2017), pp. 285--315, DOI 10.1007/s00245-016-9333-9 .

  • P. Friz, S. Gerhold, A. Pinter, Option pricing in the moderate deviations regime, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, published online on 25.8.2017, DOI 10.1111/mafi.12156 .

  • P. Friz, A. Shekhar, General rough integration, Lévy rough paths and a Lévy--Kintchine type formula, The Annals of Probability, 45 (2017), pp. 2707--2765, DOI 10.1214/16- AOP1123 .

  • P. Friz, H. Tran , On the regularity of SLE trace, Forum of Mathematics. Sigma, 5 (2017), pp. e19/1--e19/17, DOI 10.1017/fms.2017.18 .
    Abstract
    We revisit regularity of SLE trace, for all , and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia?Rodemich?Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index , improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.

  • P.K. Friz, P. Gassiat, P.-L. Lions, P.E. Souganidis, Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stochastic Partial Differential Equations. Analysis and Computations, 5 (2017), pp. 256-277, DOI 10.1007/s40072-016-0087-9 .

  • H.-J. Mucha, Assessment of stability in partitional clustering using resampling techniques, Archives of Data Science Series A, 1 (2017), pp. 21--35, DOI 10.5445/KSP/1000058747/02 .

  • V. Spokoiny, Penalized maximum likelihood estimation and effective dimension, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 389--429, DOI 10.1214/15-AIHP720 .

Contributions to Collected Editions

  • TH. Koprucki, M. Kohlhaase, D. Müller, K. Tabelow, Mathematical models as research data in numerical simulation of opto-electronic devices, in: Numerical Simulation of Optoelectronic Devices (NUSOD), 2017, pp. 225-- 226, DOI 10.1109/NUSOD.2017.8010073 .
    Abstract
    Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the stationary one-dimensional drift-diffusion equations (van Roosbroeck system).

  • M. Kohlhase, Th. Koprucki, D. Müller, K. Tabelow, Mathematical models as research data via flexiformal theory graphs, in: Intelligent Computer Mathematics: 10th International Conference, CICM 2017, Edinburgh, UK, July 17-21, 2017, Proceedings, H. Geuvers, M. England, O. Hasan, F. Rabe , O. Teschke, eds., Springer International Publishing, Cham, 2017, pp. 224--238, DOI 10.1007/978-3-319-62075-6_16 .
    Abstract
    Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open. In this paper we propose a solution -- to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexiformal framework, which has enough structure to support the various uses of models in scientific and technology workflows. Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroeck's drift-diffusion model for one-dimensional devices. This formalization -- and future extensions -- allows us to support the modeler by e.g. flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating MMS as "research data” and opens the way towards more MKM services for models.

  • CH. Bayer, H. Oberhauser, Splitting methods for SPDEs: From robustness to financial engineering, optimal control and nonlinear filtering, in: Splitting Methods in Communication, Imaging, Science, and Engineering, R. Glowinski, S.J. Osher, W. Yin, eds., Scientific Computation, Springer International Publishing Switzerland, 2017, pp. 499--539.
    Abstract
    In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control and nonlinear filtering to the viscosity theory of nonlinear SPDEs.

  • H.-J. Mucha, T.M. Gluhak, Finding Groups in Compositional Data - Some Experiments, in: Big data clustering: Data preprocessing, variable selection, and dimension reduction, WIAS Report, 2017, pp. 97--105.
    Abstract
    The talk is concerned with finding groups (clusters) in compositional data, that is nonnegative data with row sums (or column sums, respectively) equal to a constant, usually 1 in case of proportions or 100 in case of percentages. Without loss of generality, the cluster analysis of observations (row points) of compositional data is considered here, where the row profiles contains parts of some whole. Special distance functions between the profiles are proposed. Finally, applications to archaeometry are presented.

Preprints, Reports, Technical Reports

  • H.-J. Mucha, Big data clustering: Data preprocessing, variable selection, and dimension reduction, Report no. 29, WIAS, Berlin, 2017, DOI 10.20347/WIAS.REPORT.29 .
    PDF (20 MByte)

  • P. Pigato, Extreme at-the-money skew in a local volatility model, Preprint no. 2468, WIAS, Berlin, 2017.
    Abstract, PDF (270 kByte)
    We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at-the-money, we establish exact pricing formulas and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew, which explodes as T-1/2, reproducing the empirical "steep short end of the smile". This behavior does not depend on the precise choice of the parameters, but simply follows from the "regime-switch" of the local volatility at-the-money.

  • A. Lejay, P. Pigato, A threshold model for local volatility: Evidence of leverage and mean reversion effects on historical data, Preprint no. 2467, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2467 .
    Abstract, PDF (534 kByte)
    In financial markets, low prices are generally associated with high volatilities and vice-versa, this well known stylized fact usually being referred to as leverage effect. We propose a local volatility model, given by a stochastic differential equation with piecewise constant coefficients, which accounts of leverage and mean-reversion effects in the dynamics of the prices. This model exhibits a regime switch in the dynamics accordingly to a certain threshold. It can be seen as a continuous time version of the Self-Exciting Threshold Autoregressive (SETAR) model. We propose an estimation procedure for the volatility and drift coefficients as well as for the threshold level. Tests are performed on the daily prices of 21 assets. They show empirical evidence for leverage and mean-reversion effects, consistent with the results in the literature.

  • D. Belomestny, J.G.M. Schoenmakers, Regression on particle systems connected to mean-field SDEs with applications, Preprint no. 2464, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2464 .
    Abstract, PDF (268 kByte)
    In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.

  • P. Goyal, M. Redmann, Towards time-limited $H_2$-optimal model order reduction, Preprint no. 2441, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2441 .
    Abstract, PDF (363 kByte)
    In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are H_2-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited H_2-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited H_2-norm. Based on these optimality conditions, we propose an iterative scheme which upon convergences aims at satisfying these conditions. Then, we analyze how far away the obtained ROM is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to unrestricted IRKA in the time interval of interest.

  • M. Redmann, P. Kürschner, An $H_2$-type error bound for time-limited balanced truncation, Preprint no. 2440, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2440 .
    Abstract, PDF (265 kByte)
    When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an H2 error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments

  • S. Mohammadi, Ch. D'alonzo, L. Ruthotto, J. Polzehl, I. Ellerbrock, M.F. Callaghan, N. Weiskopf, K. Tabelow, Simultaneous adaptive smoothing of relaxometry and quantitative magnetization transfer mapping, Preprint no. 2432, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2432 .
    Abstract, PDF (3888 kByte)
    Attempts for in-vivo histology require a high spatial resolution that comes with the price of a decreased signal-to-noise ratio. We present a novel iterative and multi-scale smoothing method for quantitative Magnetic Resonance Imaging (MRI) data that yield proton density, apparent transverse and longitudinal relaxation, and magnetization transfer maps. The method is based on the propagation-separation approach. The adaptivity of the procedure avoids the inherent bias from blurring subtle features in the calculated maps that is common for non-adaptive smoothing approaches. The characteristics of the methods were evaluated on a high-resolution data set (500 μ isotropic) from a single subject and quantified on data from a multi-subject study. The results show that the adaptive method is able to increase the signal-to-noise ratio in the calculated quantitative maps while largely avoiding the bias that is otherwise introduced by spatially blurring values across tissue borders. As a consequence, it preserves the intensity contrast between white and gray matter and the thin cortical ribbon.

  • M. Redmann, Type II balanced truncation for deterministic bilinear control systems, Preprint no. 2425, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2425 .
    Abstract, PDF (248 kByte)
    When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an ℋ∞-error bound which so far is not known when applying type I BT to bilinear systems.

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Preprint no. 2406, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2406 .
    Abstract, PDF (450 kByte)
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1=2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • V. Avanesov, N. Buzun, Change-point detection in high-dimensional covariance structure, Preprint no. 2404, WIAS, Berlin, 2017.
    Abstract, PDF (383 kByte)
    In this paper we introduce a novel approach for an important problem of break detection. Specifically, we are interested in detection of an abrupt change in the covariance structure of a high-dimensional random process ? a problem, which has applications in many areas e.g., neuroimaging and finance. The developed approach is essentially a testing procedure involving a choice of a critical level. To that end a non-standard bootstrap scheme is proposed and theoretically justified under mild assumptions. Theoretical study features a result providing guaranties for break detection. All the theoretical results are established in a high-dimensional setting (dimensionality p  n). Multiscale nature of the approach allows for a trade-off between sensitivity of break detection and localization. The approach can be naturally employed in an on-line setting. Simulation study demonstrates that the approach matches the nominal level of false alarm probability and exhibits high power, outperforming a recent approach.

  • M. Redmann, Type II singular perturbation approximation for linear systems with Lévy noise, Preprint no. 2398, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2398 .
    Abstract, PDF (347 kByte)
    When solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems

Talks, Poster

  • A. Suvorikova, Gaussian process forecast with multidimensional distributional input, Haindorf Seminar 2018, January 23 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 25, 2018.

  • N. Buzun, Sein's method, Haindorf Seminar 2018, January 24 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2018.

  • M. Maurelli , A McKean-Vlasov SDE with reflecting boundaries, CASA Colloquium, Eindhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, January 10, 2018.

  • A. Koziuk, Gaussian comparison on a family of Eucledian balls, Haindorf Seminar 2018, January 23 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2018.

  • P. Mathé, Titel: Complexity of linear ill-posed problems in Hilbert space, Stochastisches Kolloquium, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, February 7, 2018.

  • A. Suvorikova, Construction of confidence sets in 2-Wasserstein space, Haindorf Seminar 2017, January 24 - 28, 2017, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2017.

  • A. Suvorikova, Construction of confidence sets in 2-Wasserstein space, Machine Learning Seminar, Université Paul-Sabatier, Institut de Mathématiques de Toulouse, France, December 1, 2017.

  • A. Suvorikova, Construction of non-asymptotic confidence sets in 2-Wasserstein space, Spring School ``Structural Inference'' 2017, Bad Malente, March 5 - 10, 2017.

  • A. Suvorikova, Statistical inference, WIAS-Day, Berlin, February 27 - 28, 2017.

  • N. Buzun, Bootstrap for multiple hypothesis testing, Spring School ``Structural Inference'' 2017, March 5 - 10, 2017, DFG Research Unit FOR 1735 ``Structural Inference in Statistic'', Bad Malente, March 6, 2017.

  • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 - 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

  • M. Maurelli, A McKean-Vlasov SDE with reflecting boundaries, 8th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, December 14 - 16, 2017, University of Oxford, Mathematical Institute, UK, December 15, 2017.

  • M. Maurelli, Regularization by noise for scalar conservation laws, Stochastic Analysis Day, February 25 - March 3, 2017, Universita' di Pisa, Dipartimento di Matematica, Italy, February 27, 2017.

  • M. Maurelli, Regularization by noise for scalar conservation laws, Berlin-Leipzig workshop in analysis and stochastics, November 29 - December 1, 2017, Max Planck Institute for Mathematics in the Sciences, November 30, 2017.

  • M. Maurelli, Regularization by noise for scalar conservation laws, Sé minaire de Probabililité et Statistique, Universität Nizza Sophia-Antipolis, Laboratoire Jean Alexandre Dieudonné, France, September 26, 2017.

  • M. Maurelli, Stochastic 2D Euler equations with transport noise, Chalmers University of Technology, Department of Mathematical Sciences, Gothenburg, Sweden, November 28, 2017.

  • M. Maurelli , A McKean--Vlasov SDE with reflecting boundaries, SPASS Seminar, Università di Pisa, Dipartimento di Matematica, Italy, December 18, 2017.

  • P. Pigato, Estimation of the parameters of a diffusion with discontinuous coefficients, 7th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, May 18 - 20, 2017, WIAS-Berlin, May 20, 2017.

  • P. Pigato, The Oscillating Brownian motion: estimation and application to volatility modelling, Finance and Stochastics Seminar, November 15, 2017, Imperial College London, Department of Mathematics, UK, November 15, 2017.

  • P. Pigato, The oscillating Brownian motion: estimation and application to volatility modeling, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Italy, September 26, 2017.

  • M. Redmann, A regression method to solve parabolic rough PDEs, 7th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, May 18 - 20, 2017, WIAS-Berlin, May 20, 2017.

  • M. Redmann, A regression method to solve parabolic rough PDEs, Ninth Workshop on Random Dynamical Systems, June 14 - 17, 2017, University of Bielefeld, Department of Mathematics, June 15, 2017.

  • M. Redmann, Type II singular perturbation approximation for linear systems with Levy noise, London Mathematical Society -- EPSRC Durham Symposium: Model Order Reduction, Durham University, Department of Mathematical Sciences, UK, August 14, 2017.

  • B. Stemper, Pricing in rough fractional volatility models via regularity structures, 3rd Berlin-Princeton-Singapore Workshop on Quantitative Finance, April 19 - 22, 2017, Humboldt-Universität zu Berlin, April 20, 2018.

  • CH. Bayer, A regularity structure for rough volatility, Quantitative Finance Conference in honour of Jim Gatheral's 60th Birthday, October 13 - 15, 2017, New York University, Courant Institute, USA, October 14, 2017.

  • CH. Bayer, Numerics for rough volatility models, Ninth Workshop on Random Dynamical Systems, June 14 - 17, 2017, University of Bielefeld, Department of Mathematics, June 14, 2017.

  • CH. Bayer, Rough volatility models in finance, 19th International Congress of the ÖMG and Annual DMV Meeting, September 11 - 15, 2017, Paris-Lodron University of Salzburg, Naturwissenschaftliche Fakultät, Austria, September 12, 2017.

  • CH. Bayer, Rough volatility models in finance, AMCS Seminar, King Abdullah University of Science and Technology (KAUST), Computer, Electrical and Mathematical Sciences & Engineering Division, Thuwal, Saudi Arabia, October 25, 2017.

  • CH. Bayer, Smoothing the payoff for efficient computation of basket option, Financial Math Seminar, Princeton University, Operations Research & Financial Engineering, USA, October 11, 2017.

  • CH. Bayer, Smoothing the payoff for efficient computation of basket option prices, Workshop ``Mathematics of Quantitative Finance'', February 26 - March 4, 2017, Mathematisches Forschungsinstitut Oberwolfach.

  • CH. Bayer, Smoothing the payoff for efficient computation of basket options, Workshop on Recent Developments in Numerical Methods with Applications in Statistics and Finance, June 8 - 9, 2017, University of Mannheim, Graduate School of Economics and Social Sciences, June 9, 2017.

  • CH. Bayer, Smoothing the payoff for efficient computation of basket options, Conference on Mathematical Modelling in Finance 2017, August 30 - September 2, 2017, Imperial College London, UK, September 2, 2017.

  • P. Dvurechensky, A unified view on accelerated randomized optimization methods: Coordinate descent, directional search, derivative-free method, Foundations of Computational Mathematics (FoCM 2017), Barcelona, Spain, July 17 - 19, 2017.

  • P. Dvurechensky, Adaptive similar triangles method: A stable alternative to Sinkhorn's algorithm for regularized optimal transport, Co-Evolution of Nature and Society Modelling, Problems & Experience. Devoted to Academician Nikita Moiseev centenary (Moiseev-100)., November 7 - 10, 2017, Russian Academy of Science, Federal Research Center Computer Science and Control, Moskau, Russian Federation, November 9, 2017.

  • P. Dvurechensky, Faster algorithms for optimal transport, 3. International Matheon Conference on Compressed Sensing and its Applications 2017, Berlin, December 4 - 8, 2017.

  • P. Dvurechensky, Gradient method with inexact oracle for composite non-convex optimization, Optimization and Statistical Learning, Les Houches, France, April 10 - 14, 2017.

  • P. Dvurechensky, Gradient method with inexact oracle for composite non-convex optimization, Foundations of Computational Mathematics (FoCM 2017), Barcelona, Spain, July 17 - 19, 2017.

  • P. Dvurechensky, Gradient method with inexact oracle for composite non-convex optimization, 18th French - German - Italian Conference on Optimization, September 25 - 28, 2017, Universität Paderborn, Fakultät für Elektrotechnik, Informatik und Mathematik, Paderborn, September 27, 2017.

  • P. Dvurechensky, Gradient method with inexact oracle for non convex optimization, 3rd Applied Mathematics Symposium Münster: Shape, Imaging and Optimization, February 28 - March 3, 2017.

  • P. Friz, A regularity structure for rough volatility, Global Derivates Trading & Risk Management, May 8 - 12, 2017, Barcelona, Spain, May 10, 2017.

  • P. Friz, An application of regularity structures to the analysis of rough volatility, Fractional Brownian Motion and Rough Models, June 8 - 9, 2017, Barcelona Graduate School of Economics, Spain, June 9, 2017.

  • P. Friz, Aspects of rough volatility, The 5th Imperial - ETH Workshop on Mathematical Finance, March 27 - 29, 2017, Imperial College London, Dept of Mathematics, UK, March 27, 2017.

  • P. Friz, General semimartingales and rough paths, Durham Symposium on Stochastic Analysis, July 10 - 20, 2017, Durham University, Department of Mathematical Sciences, Durham, UK, July 13, 2017.

  • P. Friz, Geometric aspects in pathwise stochastic analysis, High Risk High Gain - Groundbreaking Research in Berlin, August 31 - September 3, 2017, Technische Universität Berlin, Stabsstelle Presse, September 2, 2017.

  • P. Friz, Multiscale systems, homogenization and rough paths, Berlin-Leipzig workshop in analysis and stochastics, November 29 - December 1, 2017, Max Planck Institute for Mathematics in the Sciences, November 29, 2017.

  • P. Friz, Rough differential equations with jumps and their applications, Japanese-German Open Conference on Stochastic Analysis 2017, September 4 - 8, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, September 5, 2017.

  • A. Koziuk, Bootstrap for the regression problem with instrumental variables, Haindorf Seminar 2017, January 24 - 28, 2017, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2017.

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Statistical Foundations of Uncertainty Quantification for Inverse Problems Workshop, June 19 - 22, 2017, University of Cambridge, Center for Mathematical Sciences, UK, June 21, 2017.

  • P. Mathé, Complexity of supervised learning, ibc-paris2017 : Information Based Complexity, High-Dimensional Problems, March 14 - 15, 2017, Institut Henri Poincaré, Paris, France, March 15, 2017.

  • P. Mathé, Numerical Integration, Mini course on numerical integration, November 20 - December 4, 2017, Fudan University, School of Mathematical Sciences, China.

  • P. Mathé, Tikhonov regularization with oversmoothing penalty, 7th German Polish Conference on Optimization (GPCO 2017), August 27 - September 1, 2017, Mathematical Research and Conference Center (MRCC) of IM PAN, Będlewo, Poland, August 28, 2017.

  • H.-J. Mucha, About the influence of multiple points of bootstrapping on the validation of clustering results, European Classification and Data Analysis Conference 2017 (ECDA2017), September 27 - 29, 2017, Wrocław University of Economics, Poland, September 27, 2017.

  • H.-J. Mucha, Big data clustering: Comparison of the performance of a new fast pre-clustering and subsampling, German Polish Seminar on Data Analysis and Applications 2017, September 25 - 26, 2017, Wrocław University of Economics, Poland, September 26, 2017.

  • H.-J. Mucha, Hierarchical clustering of big data using a special bootstrapping version, AG DANK Herbsttagung, November 17 - 18, 2017, Gesellschaft für KlassifikationGesellschaft für Klassifikation, Data Science Society, November 18, 2017.

  • J. Polzehl, Connectivity networks in neuroscience - construction and analysis, Summer School 2017: Probabilistic and statistical methods for networks, August 21 - 22, 2017, Berlin Mathematical School (BMS).

  • J. Polzehl, Neue statistische Methoden zur Biomarkerselektion, Symposium: Biomarker Objektive Parameter als Grundlage für die erfolgreiche individuelle Therapie, November 21, 2017, Leibniz Gesundheitstechnologien, Berlin, November 21, 2017.

  • J. Polzehl, Structural adaptation - a statistical concept for image denoising, Variational methods and effective algorithms for imaging and vision, University of Cambridge, Isaac Newton Institute, UK, December 5, 2017.

  • J. Polzehl, Toward in-vivo histology of the brain, Neuro-Statisstics: the interface between Neuroscience, University of Minnesoata, School of Statistics (IRSA), Minneapolis, USA, May 5, 2017.

  • J. Polzehl, Towards in-vivo histology of the brain, BERLIN SYMPOSIUM 2017: Modern Statistical Methods From Data to Knowledge, December 14 - 15, 2017, Indiana Laboratory of Biostatistical Analysis of Large Data with Structure (IL-BALDS), Berlin, December 14, 2017.

  • J.G.M. Schoenmakers, Optimal stopping and control via approximative dynamic programming, Workshop on Mathematics of Deep Learning 2017, September 13 - 15, 2017, WIAS, Berlin, September 13, 2017.

  • J.G.M. Schoenmakers, Projected particle methods for solving McKean-Vlasov SDEs, Dynstoch 2017, April 5 - 7, 2017, Universität Siegen, Department Mathematik, Fachgruppe Stochastik, April 6, 2017.

  • J.G.M. Schoenmakers, Projective simulation and regression methods for Mckean-Vlasov SDE systems, Mathematisches Kolloquium, Universität Duisburg-Essen, Fakultät für Mathematik, November 29, 2017.

  • V. Spokoiny, Adaptive clustering and network clustering, 60th MIPT Scientific Conference, Moscow State University, Moscow Institute of Physics and Technology, Russian Federation, November 25, 2017.

  • V. Spokoiny, Adaptive nonparametric clustering, Optimization and Statistical Learning, Les Houches, France, April 10 - 14, 2017.

  • V. Spokoiny, Adaptive nonparametric clustering, Statistical Recovery of Discrete, Geometric and Invariant Structures, March 21 - 24, 2017, Mathematisches Forschungsinstitut Oberwolfach, March 24, 2017.

  • V. Spokoiny, Adaptive nonparametric clustering, Rencontres de Statistique Mathématique, December 18 - 22, 2017, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, December 18, 2017.

  • V. Spokoiny, Bootstrap confidence sets for spectral projectors of sample covariance (joint with A. Naumov and V.Ulyanov), Séminaire de Statistique, Université de Toulouse, Institut de Mathématiques, France, November 7, 2017.

  • V. Spokoiny, Gaussian approximation for a probability of a ball, Seminar Structural Learning, Russian Academy of Sciences, Kharkevich Institute for Information Transmission Problems, PreMoLab, Moscow, Russian Federation, June 5, 2017.

  • V. Spokoiny, Gaussian approximiation of the squared norm of a high dimensional vector, Structure Learning Seminar, Russian Academy of Sciences, Kharkevich Institute for Information Transmission Problems, PreMoLab, Russian Federation, May 18, 2017.

  • V. Spokoiny, Nonparametric estimation: parametric view, Advanced statistical methods, February 7 - 22, 2017, Independent University of Moscow, Russian Federation.

  • V. Spokoiny, Quantification of uncertainty in estimation of spectral projectors, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 6, 2017.

  • V. Spokoiny, Structural learning, Structural Learning Seminar, Russian Academy of Sciences, Institute for Information Transmission Problems, Moskau, Russian Federation, October 25, 2017.

  • V. Spokoiny, Subset selection using the smallest accepted rule, Structure Learning Seminar, Russian Academy of Sciences, Kharkevich Institute for Information Transmission Problems, PreMoLab, Moscow, Russian Federation, April 6, 2017.

  • K. Tabelow, Ch. D'alonzo, L. Ruthotto, M.F. Callaghan, N. Weiskopf, J. Polzehl, S. Mohammadi, Removing the estimation bias due to the noise floor in multi-parameter maps, The International Society for Magnetic Resonance in Medicine (ISMRM) 25th Annual Meeting & Exhibition, Honolulu, USA, April 22 - 27, 2017.

  • K. Tabelow, Ch. D'alonzo, J. Polzehl, Toward in-vivo histology of the brain, 2nd Leibniz MMs Days2017, Hannover, February 22 - 24, 2017.

  • K. Tabelow, Adaptive smoothing of Multi Parameter Maps, BERLIN SYMPOSIUM 2017: Modern Statistical Methods From Data to Knowledge, December 14 - 15, 2017, Indiana Laboratory of Biostatistical Analysis of Large Data with Structure (IL-BALDS), Berlin, December 14, 2017.

  • K. Tabelow, High resolution MRI by variance and bias reduction, IBS Channel Network Conference 2017, April 24 - 26, 2017, Hasselt Univeristy, Diepenbeek, Belgium, April 25, 2017.

  • K. Tabelow, MRI data models at low SNR, 2nd Leibniz MMs Days2017, February 22 - 24, 2017, Leibniz Informationszentrum Technik und Naturwissenschaften Technische Informationsbibliothek, Hannover, February 24, 2017, DOI 10.5446/21910 .

  • K. Tabelow, To smooth or not to smooth in fMRI, Cognitive Neuroscience Seminar, Universitätsklinikum Hamburg-Eppendorf, Institut für Computational Neuroscience, April 4, 2017.

External Preprints

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Preprint no. arXiv:1801.09540, Cornell University Library, arXiv.org, 2018.
    Abstract
    The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.

  • N. Buzun, V. Avanesov, Bootstrap for change point detection, Preprint no. arXiv:1710.07285, Cornell University Library, arXiv.org, 2017.

  • A. Bayandina, P. Dvurechensky, A. Gasnikov, F. Stonyakin, A. Titov, Mirror descent and convex optimization problems with non-smooth inequality constraints, Preprint no. arXiv:1710.06612, Cornell University Library, arXiv.org, 2017.
    Abstract
    We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. This means that neither stepsize nor stopping rule require to know the Lipschitz constant of the objective or constraint. We also construct Mirror Descent for problems with objective function, which is not Lipschitz continuous, e.g. is a quadratic function. Besides that, we address the problem of recovering the solution of the dual problem.

  • L.T. Ding, P. Mathé, Minimax rates for statistical inverse problems under general source conditions, Preprint no. arXiv:1707.01706, Cornell University Library, arXiv.org, 2017.
    Abstract
    We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon, it Minimax risk over hyperrectangles, and implications, Ann.  Statist. 18, 1990. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.

  • J. Ebert, V. Spokoiny, A. Suvorikova , Construction of non-asymptotic confidence sets in 2-Wasserstein space, Preprint no. arXiv:1703.03658, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we consider a probabilistic setting where the probability measures are considered to be random objects. We propose a procedure of construction non-asymptotic confidence sets for empirical barycenters in 2-Wasserstein space and develop the idea further to construction of a non-parametric two-sample test that is then applied to the detection of structural breaks in data with complex geometry. Both procedures mainly rely on the idea of multiplier bootstrap (Spokoiny and Zhilova (2015), Chernozhukov et al. (2014)). The main focus lies on probability measures that have commuting covariance matrices and belong to the same scatter-location family: we proof the validity of a bootstrap procedure that allows to compute confidence sets and critical values for a Wasserstein-based two-sample test.

  • B. Gess, M. Maurelli, Well-posedness by noise for scalar conservation laws, Preprint no. arXiv:1701.05393, Cornell University Library, arXiv.org, 2017.
    Abstract
    We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to well-posedness.

  • F. Götze, A. Naumov, V. Spokoiny, V. Ulyanov, Gaussian comparison and anti-concentration inequalities for norms of Gaussian random elements, Preprint no. arXiv:1708.08663, Cornell University Library, arXiv.org, 2017.
    Abstract
    We derive the bounds on the Kolmogorov distance between probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimensional-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements. We are also interested in the anti-concentration bound for a squared norm of a non-centered Gaussian element in a Hilbert space. All bounds are sharp and cannot be improved in general. We provide a list of motivation examples and applications for the derived results as well.

  • A. Naumov, V. Spokoiny, V. Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance, Preprint no. arXiv:1703.00871, Cornell University Library, arXiv.org, 2017.
    Abstract
    Let X1,?,Xn be i.i.d. sample in ?p with zero mean and the covariance matrix ?. The problem of recovering the projector onto an eigenspace of ? from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm ?Pr?P?r?2 between the true projector Pr on the subspace of the r-th eigenvalue and its empirical counterpart P?r in terms of the effective rank of ?. This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of ?Pr?P?r?2 and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

  • I. Silin, V. Spokoiny, Bayesian inference for spectral projectors of covariance matrix, Preprint no. arXiv:1711.11532, Cornell University Library, arXiv.org, 2017.

  • CH. Bayer, P. Friz, P. Gassiat, J. Martin, B. Stemper , A regularity structure for rough volatility, Preprint no. arXiv:1710.07481, Cornell University Library, arXiv.org, 2017.
    Abstract
    A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Preprint no. arXiv:1703.05132, Cornell University Library, arXiv.org, 2017.
    Abstract
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H<1/2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • P. Dvurechensky, A. Gasnikov, A. Lagunovskaya, Parallel algorithms and probability of large deviation for stochastic optimization problems, Preprint no. arXiv:1701.01830, Cornell University Library, arXiv.org, 2017.
    Abstract
    We consider convex stochastic optimization problems under different assumptions on the properties of available stochastic subgradient. It is known that, if the value of the objective function is available, one can obtain, in parallel, several independent approximate solutions in terms of the objective residual expectation. Then, choosing the solution with the minimum function value, one can control the probability of large deviation of the objective residual. On the contrary, in this short paper, we address the situation, when the value of the objective function is unavailable or is too expensive to calculate. Under "`light-tail"' assumption for stochastic subgradient and in general case with moderate large deviation probability, we show that parallelization combined with averaging gives bounds for probability of large deviation similar to a serial method. Thus, in these cases, one can benefit from parallel computations and reduce the computational time without loss in the solution quality.

  • P. Dvurechensky, A. Gasnikov, A. Tiurin, Randomized similar triangles method: A unifying framework for accelerated randomized optimization methods (Coordinate Descent, Directional Search, Derivative-Free Method), Preprint no. arXiv:1707.08486, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on accelerated random block-coordinate descent, accelerated random directional search, accelerated random derivative-free method and, using our framework, provide their versions for problems with inexact oracle information. Our contribution also includes accelerated random block-coordinate descent with inexact oracle and entropy proximal setup as well as derivative-free version of this method.

  • P. Dvurechensky, S. Omelchenko, A. Tiurin, Adaptive similar triangles method: A stable alternative to sinkhorn's algorithm for regularized optimal transport, Preprint no. arXiv:1706.07622, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we are motivated by two important applications: entropy-regularized optimal transport problem and road or IP traffic demand matrix estimation by entropy model. Both of them include solving a special type of optimization problem with linear equality constraints and objective given as a sum of an entropy regularizer and a linear function. It is known that the state-of-the-art solvers for this problem, which are based on Sinkhorn's method (also known as RSA or balancing method), can fail to work, when the entropy-regularization parameter is small. We consider the above optimization problem as a particular instance of a general strongly convex optimization problem with linear constraints. We propose a new algorithm to solve this general class of problems. Our approach is based on the transition to the dual problem. First, we introduce a new accelerated gradient method with adaptive choice of gradient's Lipschitz constant. Then, we apply this method to the dual problem and show, how to reconstruct an approximate solution to the primal problem with provable convergence rate. We prove the rate O(1/k2), k being the iteration counter, both for the absolute value of the primal objective residual and constraints infeasibility. Our method has similar to Sinkhorn's method complexity of each iteration, but is faster and more stable numerically, when the regularization parameter is small. We illustrate the advantage of our method by numerical experiments for the two mentioned applications. We show that there exists a threshold, such that, when the regularization parameter is smaller than this threshold, our method outperforms the Sinkhorn's method in terms of computation time.

  • P. Dvurechensky, Gradient method with inexact oracle for composite non-convex optimization, Preprint no. arXiv:1703.09180, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we develop new first-order method for composite non-convex minimization problems with simple constraints and inexact oracle. The objective function is given as a sum of "`hard"', possibly non-convex part, and "`simple"' convex part. Informally speaking, oracle inexactness means that, for the "`hard"' part, at any point we can approximately calculate the value of the function and construct a quadratic function, which approximately bounds this function from above. We give several examples of such inexactness: smooth non-convex functions with inexact Hölder-continuous gradient, functions given by auxiliary uniformly concave maximization problem, which can be solved only approximately. For the introduced class of problems, we propose a gradient-type method, which allows to use different proximal setup to adapt to geometry of the feasible set, adaptively chooses controlled oracle error, allows for inexact proximal mapping. We provide convergence rate for our method in terms of the norm of generalized gradient mapping and show that, in the case of inexact Hölder-continuous gradient, our method is universal with respect to Hölder parameters of the problem. Finally, in a particular case, we show that small value of the norm of generalized gradient mapping at a point means that a necessary condition of local minimum approximately holds at that point.

  • P. Mathé, B. Hofmann, Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales, Preprint no. arXiv:1705.03289, Cornell University Library, arXiv.org, 2017.