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Martin Hutzenthaler

Publications and source records attributed to Martin Hutzenthaler.

At least 19 recordsLinked to original sources
arXiv2025

Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem

Full history recursive multilevel Picard (MLP) approximations have been proved to overcome the curse of dimensionality in the numerical approximation of semilinear heat equations with nonlinearities which are globally Lipschitz continuous with respect to the maximum-norm. Nonlinearities in Hamilton-Jacobi-Bellman equations in stochastic control theory, however, are often (locally) Lipschitz continuous with respect to the standard Euclidean norm. In this paper we prove the surprising fact that MLP approximations for one such example equation suffer from the curse of dimensionality.

math.NA↗
arXiv2024

On the Itô-Alekseev-Gröbner formula for stochastic differential equations

In 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 Itô process. The introduced formula essentially generalizes both the classical Alekseev-Gröbner formula from the literature on deterministic differential equations as well as the classical Itô formula from stochastic analysis. The proposed Itô-Alekseev-Gröbner formula is a powerful tool for deriving strong approximation rates for perturbations and approximations of stochastic ordinary and partial differential equations.

math.PR↗
arXiv2024

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Recently, 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 for every p \in [1,\infty). In this article we provide sufficient conditions on the coefficient functions of the SDE and on p \in (0,\infty] which ensure local Lipschitz continuity in the strong L^p-sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear SDEs from the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

math.PR↗
arXiv2023

On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities

The classical Feynman-Kac identity represents solutions of linear partial differential equations in terms of stochastic differential euqations. This representation has been generalized to nonlinear partial differential equations on the one hand via backward stochastic differential equations and on the other hand via stochastic fixed-point equations. In this article we generalize the representation via stochastic fixed-point equations to allow the nonlinearity in the semilinear partial differential equation to depend also on the gradient of the solution.

math.PR↗
arXiv2023

On existence and uniqueness properties for solutions of stochastic fixed point equations with gradient-dependent nonlinearities

The combination of the Itô formula and the Bismut-Elworthy-Li formula implies that suitable smooth solutions of semilinear Kolmogorov partial differential equations (PDEs) are also solutions to certain stochastic fixed point equations (SFPEs). In this paper we generalize known results on existence and uniqueness of solutions of SFPEs associated with PDEs with Lipschitz continuous, gradient-independent nonlinearities to the case of gradient-dependent nonlinearities. The main challenge arises from the fact that in the case of a non-differentiable terminal condition and a gradient-dependent nonlinearity the Bismut-Elworthy-Li formula leads to a singularity of the solution of the SFPE in the last time point.

math.PR↗
arXiv2023

Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions

In many numerical simulations stochastic gradient descent (SGD) type optimization methods perform very effectively in the training of deep neural networks (DNNs) but till this day it remains an open problem of research to provide a mathematical convergence analysis which rigorously explains the success of SGD type optimization methods in the training of DNNs. In this work we study SGD type optimization methods in the training of fully-connected feedforward DNNs with rectified linear unit (ReLU) activation. We first establish general regularity properties for the risk functions and their generalized gradient functions appearing in the training of such DNNs and, thereafter, we investigate the plain vanilla SGD optimization method in the training of such DNNs under the assumption that the target function under consideration is a constant function. Specifically, we prove under the assumption that the learning rates (the step sizes of the SGD optimization method) are sufficiently small but not $L^1$-summable and under the assumption that the target function is a constant function that the expectation of the riskof the considered SGD process converges in the training of such DNNs to zero as the number of SGD steps increases to infinity.

cs.LG↗
arXiv2022

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

math.NA↗
arXiv2022

Costly defense traits in structured populations

We propose a model for the dynamics of frequencies of a costly defense trait. More precisely, we consider Lotka-Volterra-type models involving a prey (or host) population consisting of two types and a predator (or parasite) population, where one type of prey individuals - modeling carriers of a defense trait - is more effective in defending against the predators but has a weak reproductive disadvantage. Under certain assumptions we prove that the relative frequency of these defenders in the total prey population converges to spatially structured Wright-Fisher diffusions with frequency-dependent migration rates. For the many-demes limit (mean-field approximation) hereof, we show that the defense trait goes to fixation/extinction if and only if the selective disadvantage is smaller/larger than an explicit function of the ecological model parameters.

math.PR↗
arXiv2022

A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations

We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to Itô processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.

math.PR↗
arXiv2022

Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven in the case of semilinear heat equations.

math.NA↗
arXiv2022

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.

math.NA↗
arXiv2022

On moments and strong local Hölder regularity of solutions of stochastic differential equations and of their spatial derivative processes

Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature. Counterexamples with smooth and bounded coefficients demonstrate that the non-globally Lipschitz case is more subtle. In this article we establish conditions, including a suitable local monotonicity property, which provide existence of continuously differentiable solutions of SDEs, moment estimates and strong local Hölder regularity.

math.PR↗
arXiv2022

Stopped Brownian-increment tamed Euler method

In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.

math.PR↗
arXiv2022

On the speed of convergence of Picard iterations of backward stochastic differential equations

It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is $z$-independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations.

math.PR↗
arXiv2022

Multilevel Picard approximations for high-dimensional decoupled forward-backward stochastic differential equations

Backward stochastic differential equations (BSDEs) appear in numeruous applications. Classical approximation methods suffer from the curse of dimensionality and deep learning-based approximation methods are not known to converge to the BSDE solution. Recently, Hutzenthaler et al. (arXiv:2108.10602) introduced a new approximation method for BSDEs whose forward diffusion is Brownian motion and proved that this method converges with essentially optimal rate without suffering from the curse of dimensionality. The central object of this article is to extend this result to general forward diffusions. The main challenge is that we need to establish convergence in temporal-spatial Hölder norms since the forward diffusion cannot be sampled exactly in general.

math.PR↗
arXiv2022

The Kolmogorov backward equation for stochastic Burgers equations and for stochastic 2D-Navier-Stokes equations

In this book we establish under suitable assumptions the uniqueness and existence of viscosity solutions of Kolmogorov backward equations for stochastic partial differential equations (SPDEs). In addition, we show that this solution is the semigroup of the corresponding SPDE. This generalizes the Feynman-Kac formula to SPDEs and establishes a link between solutions of Kolmogorov equations and solutions of the corresponding SPDEs. In contrast to the literature we only assume that the nonlinear part of the drift is Lipschitz continuous on bounded sets (and not globally Lipschitz continuous) and we allow the diffusion coefficient to be degenerate and non-constant. In the last part of this book we apply our results to stochastic Burgers equations and to stochastic 2-D Navier-Stokes equations.

math.PR↗
arXiv2022

Strong convergence rate of Euler-Maruyama approximations in temporal-spatial Hölder-norms

Classical approximation results for stochastic differential equations analyze the $L^p$-distance between the exact solution and its Euler-Maruyama approximations. In this article we measure the error with temporal-spatial Hölder-norms. Our motivation for this are multigrid approximations of the exact solution viewed as a function of the starting point. We establish the classical strong convergence rate $0.5$ with respect to temporal-spatial Hölder-norms if the coefficient functions have bounded derivatives of first and second order.

math.NA↗