#### Prof. Dr. Martin Hutzenthaler

Stochastic Analysis

University of Duisburg-Essen

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- martin.hutzenthaler@uni-due.de
- +49 201 183 6844
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**Multilevel picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities**

Hutzenthaler, M. and Kruse, T.*SIAM Journal on Numerical Analysis*58 (2020)Parabolic partial differential equations (PDEs) and backward stochastic differential equations have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy. © 2020 Martin Hutzenthaler and Thomas Kruseview abstract 10.1137/17M1157015 **Numerical simulations for full history recursive multilevel picard approximations for systems of high-dimensional partial differential equations**

Becker, S. and Braunwarth, R. and Hutzenthaler, M. and Jentzen, A. and von Wurstemberger, P.*Communications in Computational Physics*28 (2020)One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very hard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by the algorithm to compute an approximation of accuracy ε>0 grows at most polynomially in both the reciprocal 1/ε of the required accuracy and the dimension d∈N of the PDE. Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons. It is a key contribution of this article to extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposed MLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs, systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to 1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientific literature. © 2020 Global-Science Pressview abstract 10.4208/CICP.OA-2020-0130 **On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients**

Hutzenthaler, M. and Jentzen, A.*Annals of Probability*48 (2020)We develop 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 Lp-distance between the solution process of an SDE and an arbitrary Ito process, which we view as a perturbation of the solution process of the SDE, by the Lp-distances of the differences of the local characteristics for suitable p, q > 0. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn-Hilliard-Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations. © Institute of Mathematical Statistics, 2020.view abstract 10.1214/19-AOP1345 **Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks**

Hutzenthaler, M. and Jentzen, A. and Wurstemberger, P.V.*Electronic Journal of Probability*25 (2020)Parabolic 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 approximately determine fair prices of financial derivatives in the financial engineering industry. The PDEs appearing in financial engineering applications are often nonlinear (e.g., in PDE models which take into account the possibility of a defaulting counterparty) and high-dimensional since the dimension typically corresponds to the number of considered financial assets. A major issue in the scientific literature is that most approximation methods for nonlinear PDEs suffer from the so-called curse of dimensionality in the sense that the computational effort to compute an approximation with a prescribed accuracy grows exponentially in the dimension of the PDE or in the reciprocal of the prescribed approximation accuracy and nearly all approximation methods for nonlinear PDEs in the scientific literature have not been shown not to suffer from the curse of dimensionality. Recently, a new class of approximation schemes for semilinear parabolic PDEs, termed full history recursive multilevel Picard (MLP) algorithms, were introduced and it was proven that MLP algorithms do overcome the curse of dimensionality for semilinear heat equations. In this paper we extend and generalize those findings to a more general class of semilinear PDEs which includes as special cases the important examples of semilinear Black-Scholes equations used in pricing models for financial derivatives with default risks. In particular, we introduce an MLP algorithm for the approximation of solutions of semilinear Black-Scholes equations and prove, under the assumption that the nonlinearity in the PDE is globally Lipschitz continuous, that the computational effort of the proposed method grows at most polynomially in both the dimension and the reciprocal of the prescribed approximation accuracy. We thereby establish, for the first time, that the numerical approximation of solutions of semilinear Black-Scholes equations is a polynomially tractable approximation problem. © 2020, Institute of Mathematical Statistics. All rights reserved.view abstract 10.1214/20-EJP423 **Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations**

Beck, C. and Hornung, F. and Hutzenthaler, M. and Jentzen, A. and Kruse, T.*Journal of Numerical Mathematics*28 (2020)One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen-Cahn PDEs) by introducing and analyzing truncated variants of the recently introduced full-history recursive multilevel Picard approximation schemes. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.view abstract 10.1515/jnma-2019-0074 **Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations: Overcoming the curse of dimensionality**

Hutzenthaler, M. and Jentzen, A. and Kruse, T. and Anh Nguyen, T. and Von Wurstemberger, P.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*476 (2020)For a long time it has been 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 the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy. © 2020 The Author(s).view abstract 10.1098/rspa.2019.0630rspa20190630 **Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime**

HUTZENTHALER, M. and PIEPER, D.*Annals of Applied Probability*30 (2020)Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are "of the same size". In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this "sparse regime"the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees. © Institute of Mathematical Statistics, 2020.view abstract 10.1214/20-AAP1559 **On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations**

Weinan, E. and Hutzenthaler, M. and Jentzen, A. and Kruse, T.*Journal of Scientific Computing*79 (2019)Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in pricing and hedging models for financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article (E et al., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, arXiv:1607.03295) we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution of a semilinear heat equation that the computational complexity is bounded by O(dε-(4+δ)) for any δ∈ (0 , ∞) where d is the dimensionality of the problem and ε∈ (0 , ∞) is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for many of these 100-dimensional example PDEs are very satisfactory in terms of both accuracy and speed. Moreover, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the scientific literature. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.view abstract 10.1007/s10915-018-00903-0 **Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations**

Hutzenthaler, M. and Jentzen, A. and Wang, X.*Mathematics of Computation*87 (2018)Exponential 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 approximation processes such as Euler- Maruyama approximations, linear-implicit Euler approximations, and some tamed Euler approximations from the literature rarely preserve exponential integrability properties of the exact solution. The main contribution of this article is to identify a class of stopped increment-tamed Euler approximations which preserve exponential integrability properties of the exact solution under minor additional assumptions on the involved functions. © 2017 American Mathematical Society.view abstract 10.1090/mcom/3146 **Loss of regularity for Kolmogorov equations**

Hairer, M. and Hutzenthaler, M. and Jentzen, A.*Annals of Probability*43 (2015)The celebrated Hörmander 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 PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander's condition even if the initial function is only continuous but not differentiable. Firstorder linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Hölder continuous. From the perspective of probability theory, the existence of this example PDE has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally Hölder continuous. In other words, degenerate noise can have a roughening effect. A further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coefficients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense, but at a rate that is slower then any power law. © Institute of Mathematical Statistics, 2015.view abstract 10.1214/13-AOP838 **Numerical approximations of stochastic differential equations with non-globally lipschitz continuous coefficients**

Hutzenthaler, M. and Jentzen, A.*Memoirs of the American Mathematical Society*236 (2015)Many 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 these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry. © 2014 American Mathematical Society.view abstract 10.1090/memo/1112 **Divergence of the multilevel Monte Carlo Euler method for nonlinear stochastic differential equations**

Hutzenthaler, M. and Jentzen, A. and Kloeden, P.E.*Annals of Applied Probability*23 (2013)The 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 Carlo simulations do, however, not suffer from this divergence behavior of Euler's method because this divergence behavior happens on rare events. Indeed, for such nonlinear SDEs the classical Monte Carlo Euler method has been shown to converge by exploiting that the Euler approximations diverge only on events whose probabilities decay to zero very rapidly. Significantly more efficient than the classical Monte Carlo Euler method is the recently introduced multilevel Monte Carlo Euler method. The main observation of this article is that this multilevel Monte Carlo Euler method does-in contrast to classical Monte Carlo methods-not converge in general in the case of such nonlinear SDEs. More precisely, we establish divergence of the multilevel Monte Carlo Euler method for a family of SDEs with superlinearly growing and globally one-sided Lipschitz continuous drift coefficients. In particular, the multilevelMonte Carlo Euler method diverges for these nonlinear SDEs on an event that is not at all rare but has probability one. As a consequence for applications, we recommend not to use the multilevel Monte Carlo Euler method for SDEs with superlinearly growing nonlinearities. Instead we propose to combine the multilevelMonte Carlo method with a slightly modified Euler method. More precisely, we show that the multilevel Monte Carlo method combined with a tamed Euler method converges for nonlinear SDEs with globally one-sided Lipschitz continuous drift coefficients and preserves its strikingly higher order convergence rate from the Lipschitz case. © Institute of Mathematical Statistics, 2013.view abstract 10.1214/12-AAP890 **Ecological and genetic effects of introduced species on their native competitors**

Wittmann, M.J. and Hutzenthaler, M. and Gabriel, W. and Metzler, D.*Theoretical Population Biology*84 (2013)Species introductions to new habitats can cause a decline in the population size of competing native species and consequently also in their genetic diversity. We are interested in why these adverse effects are weak in some cases whereas in others the native species declines to the point of extinction. While the introduction rate and the growth rate of the introduced species in the new environment clearly have a positive relationship with invasion success and impact, the influence of competition is poorly understood. Here, we investigate how the intensity of interspecific competition influences the persistence time of a native species in the face of repeated and ongoing introductions of the nonnative species. We analyze two stochastic models: a model for the population dynamics of both species and a model that additionally includes the population genetics of the native species at a locus involved in its adaptation to a changing environment. Counterintuitively, both models predict that the persistence time of the native species is lowest for an intermediate intensity of competition. This phenomenon results from the opposing effects of competition at different stages of the invasion process: With increasing competition intensity more introduction events are needed until a new species can establish, but increasing competition also speeds up the exclusion of the native species by an established nonnative competitor. By comparing the ecological and the eco-genetic model, we detect and quantify a synergistic feedback between ecological and genetic effects. © 2012 Elsevier Inc.view abstract 10.1016/j.tpb.2012.11.003 **Interacting diffusions and trees of excursions: Convergence and comparison**

Hutzenthaler, M.*Electronic Journal of Probability*17 (2012)We consider systems of interacting diffusions with local population regulation representing populations on countably many islands. Our main result shows that the total mass process of such a system is bounded above by the total mass process of a tree of excursions with appropriate drift and diffusion coefficients. As a corollary, this entails a sufficient, explicit condition for extinction of the total mass as time tends to infinity. On the way to our comparison result, we establish that systems of interacting diffusions with uniform migration between finitely many islands converge to a tree of excursions as the number of islands tends to infinity. In the special case of logistic branching, this leads to a duality between a tree of excursions and the solution of a McKean-Vlasov equation.view abstract 10.1214/EJP.v17-2278 **Norovirus GII.4 and GII.7 capsid sequences undergo positive selection in chronically infected patients**

Hoffmann, D. and Hutzenthaler, M. and Seebach, J. and Panning, M. and Umgelter, A. and Menzel, H. and Protzer, U. and Metzler, D.*Infection, Genetics and Evolution*12 (2012)Norovirus has become an important cause for infectious gastroenteritis. Particularly genotype II.4 (GII.4) has been shown to spread rapidly and causes worldwide pandemics. Emerging new strains evade population immunity and lead to high norovirus prevalence. Chronic infections have been described recently and will become more prevalent with increasing numbers of immunocompromized patients. Here, we studied norovirus evolution in three chronically infected patients, two genotypes II.4 and one II.7. A 719 and 757. nt region was analyzed for GII.4 and GII.7, respectively. This covers the entire hypervariable P2 domain of the VP1 capsid gene. Genetic variability at given and between different time points was assessed. Evolutionary adaptation was analyzed by Bayesian sampling of genealogies. This analysis clearly demonstrated positive selection rather than incidental drift for all three strains. The GII.7 and one GII.4 strain accumulated on average 5-9 mutations per 100. days, most of them non-synonymous. This is a much higher evolutionary rate than observed for noroviruses on a global level.Our data demonstrate that norovirus quasispecies are positively selected in chronically infected patients. The numbers of intraindividual amino acid mutations acquired in the capsid gene are similar to those separating consecutive GII.4 epidemic strains. Evolution in a given, chronically infected individual may thus generate novel genotypes at risk to expedite global evolution particularly for slowly evolving genotypes, as GII.7. © 2012 Elsevier B.V.view abstract 10.1016/j.meegid.2012.01.020 **Strong convergence of an explicit numerical method for sdes with nonglobally lipschitz continuous coefficients**

Hutzenthaler, M. and Jentzen, A. and Kloeden, P.E.*Annals of Applied Probability*22 (2012)On 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 implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme. © 2012 Institute of Mathematical Statistics.view abstract 10.1214/11-AAP803 **Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients**

Hutzenthaler, M. and Jentzen, A.*Foundations of Computational Mathematics*11 (2011)Stochastic 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. However, the important case of superlinearly growing coefficients has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth. © 2011 SFoCM.view abstract 10.1007/s10208-011-9101-9 **Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients**

Hutzenthaler, M. and Jentzen, A. and Kloeden, P.E.*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*467 (2011)The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler's approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense. © 2010 The Royal Society.view abstract 10.1098/rspa.2010.0348 **Time reversal of some stationary jump diffusion processes from population genetics**

Hutzenthaler, M. and Taylor, J.E.*Advances in Applied Probability*42 (2010)We describe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives each bottleneck, the forward process is a diffusion on [0, 1] that jumps to the boundary before diffusing back into the interior. We showthat the behavior of the time-reversed process depends on whether the boundaries are accessible to the diffusive motion of the forward process. If a boundary point is inaccessible to the forward diffusion then time reversal leads to a jump diffusion that jumps immediately into the interior whenever it arrives at that point. If, instead, a boundary point is accessible then the jumps off of that point are governed by a weighted local time of the time-reversed process. © Applied Probability Trust 2010.view abstract 10.1239/aap/1293113155

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