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Taylor expansions of solutions of stochastic partial differential equationsApr 15 2009The solutions of parabolic and hyperbolic stochastic partial differential equations (SPDEs) driven by an infinite dimensional Brownian motion, which is a martingale, are in general not semi-martingales any more and therefore do not satisfy an It\^o formula ... More

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficientsJan 01 2014We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $ L^p $-distance ... More

Taylor expansions of solutions of stochastic partial differential equations with additive noiseOct 01 2010The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an It\^{o} formula like the solution of a finite-dimensional ... More

Convergence of the stochastic Euler scheme for locally Lipschitz coefficientsDec 14 2009Nov 17 2011Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case ... More

Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equationsJan 21 2016This article proposes and analyzes explicit and easily implementable temporal numerical approximation schemes for additive noise-driven stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as, e.g., stochastic Ginzburg-Landau ... More

Exponential moments for numerical approximations of stochastic partial differential equationsSep 22 2016Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs ... More

Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficientsMar 26 2012May 09 2013Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for ... More

Galerkin approximations for the stochastic Burgers equationApr 11 2013Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic ... More

Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equationsMay 02 2011Sep 10 2013The Euler-Maruyama scheme is known to diverge strongly and numerically weakly when applied to nonlinear stochastic differential equations (SDEs) with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. Classical Monte ... More

A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientsSep 19 2018In recent years deep artificial neural networks (DNNs) have very successfully been employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, ... More

On the Alekseev-Gröbner formula in Banach spacesOct 23 2018The Alekseev-Gr\"obner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gr\"obner formula ... More

Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial valuesDec 21 2015Oct 06 2016In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. ... More

Loss of regularity for Kolmogorov equationsSep 26 2012Mar 06 2015The celebrated H\"{o}rmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov ... More

Efficient simulation of nonlinear parabolic SPDEs with additive noiseOct 31 2012Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649-667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The ... More

Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationsSep 18 2017High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications ... More

Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equationsSep 29 2013Jan 06 2014Exponential integrability properties of numerical approximations are a key tool for establishing positive rates of strong and numerically weak convergence for a large class of nonlinear stochastic differential equations. It turns out that well-known numerical ... More

Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficientsMay 04 2009Jul 05 2011The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz continuous drift and diffusion coefficient. Recent results extend this convergence to coefficients which grow at most ... More

An exponential Wagner-Platen type scheme for SPDEsSep 18 2013The strong numerical approximation of semilinear stochastic partial differential equations (SPDEs) driven by infinite dimensional Wiener processes is investigated. There are a number of results in the literature that show that Euler-type approximation ... More

On stochastic differential equations with arbitrary slow convergence rates for strong approximationJun 09 2015In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable ... More

Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equationsSep 29 2013Nov 20 2016Exponential integrability properties of numerical approximations are a key tool for establishing positive rates of strong and numerically weak convergence for a large class of nonlinear stochastic differential equations. It turns out that well-known numerical ... More

Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationsJun 15 2017We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the ... More

Deep optimal stoppingApr 15 2018Jan 29 2019We develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test ... More

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equationsSep 22 2013Nov 24 2014Recently, Hairer et. al (2012) showed that there exist SDEs with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L^p-sense with respect to the initial value ... More

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationsSep 07 2018Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational ... More

On the Itô-Alekseev-Gröbner formula for stochastic differential equationsDec 24 2018In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an It\^o process. The introduced formula essentially generalizes both the classical Alekseev-Gr\"obner ... More

Solving stochastic differential equations and Kolmogorov equations by means of deep learningJun 01 2018Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, ... More

On the differentiability of solutions of stochastic evolution equations with respect to their initial valuesNov 03 2016In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients ... More

DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option PricingSep 20 2018Sep 26 2018We analyze approximation rates by deep ReLU networks of a class of multi-variate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReLU architectures capable of efficiently approximating tensor products. Combining ... More

Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equationsNov 01 2018Optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise are obtained. In particular, we establish the optimality of strong convergence rates for full-discrete ... More

Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equationsJan 16 2017Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical approximation ... More

On the mild Itô formula in Banach spacesDec 09 2016The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& R\"ockner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study solutions ... More

Strong error analysis for stochastic gradient descent optimization algorithmsJan 29 2018Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every ... More

Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noiseAug 21 2015Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be solved explicitly, ... More

Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noiseMay 21 2010Nov 04 2011In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.

A Milstein scheme for SPDEsJan 15 2010Jan 27 2012This article studies an infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs). The Milstein scheme is known to be impressively efficient for SODEs which fulfill a certain commutativity ... More

On arbitrarily slow convergence rates for strong numerical approximations of Cox-Ingersoll-Ross processes and squared Bessel processesFeb 28 2017Nov 06 2017Cox-Ingersoll-Ross (CIR) processes are extensively used in state-of-the-art models for the approximative pricing of financial derivatives. In particular, CIR processes are day after day employed to model instantaneous variances (squared volatilities) ... More

Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearitiesApr 14 2015In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. We establish strong convergence rates for ... More

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensionsMay 03 2016We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process which is strongly H\"older continuous in time, then this sequence converges in the strong sense even with respect ... More

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensionsMay 03 2016Mar 16 2017We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process which is strongly H\"older continuous in time, then this sequence converges in the strong sense even with respect ... More

Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearitiesMar 14 2019The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing ... More

A mild Ito formula for SPDEsSep 18 2010Jul 25 2012This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical ... More

Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficientsOct 18 2010Sep 12 2012On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the ... More

Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationsSep 09 2018The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined Deep Learning, revolutionized the area of artificial intelligence, machine learning, and data ... More

Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundariesMar 25 2014Cox-Ingersoll-Ross (CIR) processes are widely used in financial modeling such as in the Heston model for the approximative pricing of financial derivatives. Moreover, CIR processes are mathematically interesting due to the irregular square root function ... More

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risksMar 14 2019Parabolic partial differential equations (PDEs) are widely used in the mathematical modeling of natural phenomena and man made complex systems. In particular, parabolic PDEs are a fundamental tool to determine fair prices of financial derivatives in the ... More

Counterexamples to regularities for the derivative processes associated to stochastic evolution equationsMar 27 2017In the recent years there has been an increased interest in studying regularity properties of the derivatives of stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has been shown ... More

Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equationsOct 19 2017In this paper we propose and analyze explicit space-time discrete numerical approximations for additive space-time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers ... More

Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutionsDec 12 2018In this article we establish exponential moment bounds, moment bounds in fractional order smoothness spaces, a uniform H\"older continuity in time, and strong convergence rates for a class of fully discrete exponential Euler-type numerical approximations ... More

Weak convergence rates for temporal numerical approximations of stochastic wave equations with multiplicative noiseJan 16 2019In numerical analysis for stochastic partial differential equations one distinguishes between weak and strong convergence rates. Often the weak convergence rate is twice the strong convergence rate. However, there is no standard way to prove this: to ... More

Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equationsApr 07 2016May 17 2016This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly ... More

Renormalized powers of Ornstein-Uhlenbeck processes and well-posedness of stochastic Ginzburg-Landau equationsFeb 24 2013Apr 11 2013This article analyzes well-definedness and regularity of renormalized powers of Ornstein-Uhlenbeck processes and uses this analysis to establish local existence, uniqueness and regularity of strong solutions of stochastic Ginzburg-Landau equations with ... More

On full history recursive multilevel Picard approximations and numerical approximations of high-dimensional nonlinear parabolic partial differential equationsJul 12 2016Parabolic partial differential equations (PDEs) are a fundamental tool in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high-dimensional and nonlinear. Since ... More

Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spacesNov 03 2016In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov ... More

Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equationsNov 07 2017The scientific literature contains a number of numerical approximation results for stochastic partial differential equations (SPDEs) with superlinearly growing nonlinearities but, to the best of our knowledge, none of them prove strong or weak convergence ... More

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equationsJan 30 2019Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural ... More

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equationsJul 03 2018For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of ... More

Status and Prospects of Top-Quark PhysicsApr 16 2009Jul 31 2009The top quark is the heaviest elementary particle observed to date. Its large mass of about 173 GeV/c^2 makes the top quark act differently than other elementary fermions, as it decays before it hadronises, passing its spin information on to its decay ... More

Zinc Electrode Shape-Change in Secondary Air Batteries: A 2D Modeling ApproachMar 01 2019Zinc-air batteries offer large specific energy densities, while relying on abundant and non-toxic materials. In this paper, we present the first multi-dimensional simulations of zinc-air batteries. We refine our existing theory-based model of secondary ... More

Frequency dependent specific heat of viscous silicaAug 01 2000We apply the Mori-Zwanzig projection operator formalism to obtain an expression for the frequency dependent specific heat c(z) of a liquid. By using an exact transformation formula due to Lebowitz et al., we derive a relation between c(z) and K(t), the ... More

Lifshitz transitions and elastic properties of Osmium under pressureJul 13 2006Apr 19 2007Topological changes of the Fermi surface under pressure may cause anomalies in the low-temperature elastic properties. Our density functional calculations for elemental Osmium evidence that this metal undergoes three such Lifshitz transitions in the pressure ... More

Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases II: Open SystemsAug 21 2000We calculate the spectrum of Lyapunov exponents for a point particle moving in a random array of fixed hard disk or hard sphere scatterers, i.e. the disordered Lorentz gas, in a generic nonequilibrium situation. In a large system which is finite in at ... More

B. B. G. K. Y. Hierarchy Methods for Sums of Lyapunov Exponents for Dilute GasesJan 12 1998We consider a general method for computing the sum of positive Lyapunov exponents for moderately dense gases. This method is based upon hierarchy techniques used previously to derive the generalized Boltzmann equation for the time dependent spatial and ... More

Characterization of gas diffusion electrodes for metal-air batteriesJan 30 2017Gas diffusion electrodes are commonly used in high energy density metal-air batteries for the supply of oxygen. Hydrophobic binder materials ensure the coexistence of gas and liquid phase in the pore network. The phase distribution has a strong influence ... More

A likelihood-based reconstruction algorithm for top-quark pairs and the KLFitter frameworkDec 19 2013Mar 12 2014A likelihood-based reconstruction algorithm for arbitrary event topologies is introduced and, as an example, applied to the single-lepton decay mode of top-quark pair production. The algorithm comes with several options which further improve its performance, ... More

How good is the generalized Langevin equation to describe the dynamics of photo-induced electron transfer in fluid solution?Apr 20 2017The dynamics of unimolecular photo-triggered reactions can be strongly affected by the surrounding medium. An accurate description of these reactions requires knowing the free energy surface (FES) and the friction felt by the reactants. Most of theories ... More

Electronic Structure of an Iron-Porphyrin Derivative on Au(111)Nov 25 2018Surface-bound porphyrins are promising candidates for molecular switches, electronics and spintronics. Here, we studied the structural and the electronic properties of Fe-tetra-pyridil-porphyrin adsorbed on Au(111) in the monolayer regime. We combined ... More

Towards Rechargeable Zinc-Air Batteries with Aqueous Chloride ElectrolytesFeb 12 2019This paper presents a combined theoretical and experimental investigation of aqueous near-neutral electrolytes based on chloride salts for rechargeable zinc-air batteries (ZABs). The resilience of near-neutral chloride electrolytes in air could extend ... More

MULTIBAT: Unified workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reductionApr 13 2017We present a simulation workflow for efficient investigations of the interplay between 3D lithium-ion electrode microstructures and electrochemical performance, with emphasis on lithium plating. Our approach addresses several challenges. First, the 3D ... More

MULTIBAT: Unified workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reductionApr 13 2017Mar 08 2018We present a simulation workflow for efficient investigations of the interplay between 3D lithium-ion electrode microstructures and electrochemical performance, with emphasis on lithium plating. Our approach addresses several challenges. First, the 3D ... More

Characterization of Dimethylsulfoxide / Glycerol Mixtures: A Binary Solvent System for the Study of "Friction-Dependent" Chemical ReactivityMar 04 2016The properties of binary mixtures of dimethylsulfoxide and glycerol, measured by several techniques, are reported. Special attention is given to those properties contributing or affecting chemical reactions. In this respect the investigated mixture behaves ... More

Inter-Domain Charge Transfer as a Rationale for Superior Photovoltaic Performances of Mixed Halide Lead PerovskitesAug 08 2018Organic-inorganic lead halide perovskites containing a mixture of iodide and bromide anions consistently perform better in donor-acceptor heterojunction solar cells than the standard methylammonium lead triiodide material. This observation is counterintuitive, ... More