arXiv subjects

Thomas Müller-Gronbach

Publications and source records attributed to Thomas Müller-Gronbach.

13 recordsLinked to original sources

arXiv2026

On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient

We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient. More precisely, we essentially assume that the drift coefficient is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$ that is an orientable $C^4$-hypersurface of positive reach, the diffusion coefficient is assumed to be Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and the diffusion coefficient has a non-degenerate portion orthogonal to $Θ$. In recent years, a number of results have been proven in the literature for strong approximation of such SDEs and, in particular, the performance of the Euler-Maruyama scheme was studied. For $d=1$ and finite $Θ$ it was shown that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\geq 1$ as in the classical case of Lipschitz continuous coefficients. For $d>1$, it was only known so far, that the Euler-Maruyama scheme achieves an $L_2$-error rate of at least $1/4-$ if, additionally, the coefficients $μ$ and $σ$ are globally bounded. In this article, we prove that in the above setting the Euler-Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2-$ for all $d\in\mathbb{N}$ and all $p\geq 1$. The proof of this result is based on the well-known approach of transforming such an SDE into an SDE with globally Lipschitz continuous coefficients, a new Itô formula for a class of functions which are not globally $C^2$ and a detailed analysis of the expected total time that the actual position of the time-continuous Euler-Maruyama scheme and its position at the preceding time point on the underlying grid are on 'different sides' of the hypersurface $Θ$.

math.NA↗

arXiv2026

Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of Hölder or Sobolev regularity using a Weierstraß scale

We study strong approximation of solutions of SDEs with bounded $α$-Hölder continuous drift coefficient and constant diffusion coefficient at time point $1$. Recently, it was shown in [arXiv:1909.07961v4 (2021)] that for such SDEs the equidistant Euler scheme achieves an $L^p$-error rate of at least $(1+α)/2$, up to an arbitrary small $\varepsilon$, for all $p\geq 1$ and $α\in (0,1]$, in terms of the number of evaluations of the driving Brownian motion $W$. In this article, we prove a matching lower error bound for $α\in (0,1)$. More precisely, we show that for every $α\in (0,1)$, the $L^p$-error rate $(1+α)/2$ of the Euler scheme in [arXiv:1909.07961v4 (2021)] cannot be improved in general by any numerical method based on finitely many evaluations of $W$ in $[0,1]$. Up to now, this result was known only for $α=1$. Even stronger, an $L^p$-error rate better than $(1+α)/2$ cannot be achieved, even if algorithms additionally use a finite number of time integrals of $W$. Thus, Wagner-Platen type schemes are not superior to the Euler scheme. Additionally, we extend a result from [arXiv:2402.13732v2 (2024)] on final time approximation of SDEs with a bounded drift coefficient of fractional Sobolev regularity $α\in (0,1)$. We prove that for every $α\in (0,1)$, the $L^p$-error rate $(1+ α)/2$ shown in [arXiv:2101.12185v2 (2022)] for the equidistant Euler scheme can essentially not be improved by any numerical method based on finitely many evaluations and time integrals of $W$ in $[0,1]$. This lower bound was known from [arXiv:2402.13732v2 (2024)] only for $α\in (1/2,1)$, $p=2$ and numerical methods based on finitely many evaluations of $W$. For the proof of our results we use variants of the Weierstrass function as a drift coefficient and we extend the coupling of noise technique introduced in [arXiv:2010.00915v1 (2020)].

math.PR↗

arXiv2024

On optimal error rates for strong approximation of SDEs with a drift coefficient of fractional Sobolev regularity

We study strong approximation of scalar additive noise driven stochastic differential equations (SDEs) at time point $1$ in the case that the drift coefficient is bounded and has Sobolev regularity $s\in(0,1)$. Recently, it has been shown in [arXiv:2101.12185v2 (2022)] that for such SDEs the equidistant Euler approximation achieves an $L^2$-error rate of at least $(1+s)/2$, up to an arbitrary small $\varepsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In the present article we prove a matching lower error bound for $s\in(1/2,1)$. More precisely we show that, for every $s\in(1/2,1)$, the $L^2$-error rate $(1+s)/2$ can, up to a logarithmic term, not be improved in general by no numerical method based on finitely many evaluations of $W$ at fixed time points. Up to now, this result was known in the literature only for the cases $s=1/2-$ and $s=1-$. For the proof we employ the coupling of noise technique recently introduced in [arXiv:2010.00915 (2020)] to bound the $L^2$-error of an arbitrary approximation from below by the $L^2$-distance of two occupation time functionals provided by a specifically chosen drift coefficient with Sobolev regularity $s$ and two solutions of the corresponding SDE with coupled driving Brownian motions. For the analysis of the latter distance we employ a transformation of the original SDE to overcome the problem of correlated increments of the difference of the two coupled solutions, occupation time estimates to cope with the lack of regularity of the chosen drift coefficient around the point $0$ and scaling properties of the drift coefficient.

math.PR↗

arXiv2022

Existence, uniqueness and approximation of solutions of SDEs with superlinear coefficients in the presence of discontinuities of the drift coefficient

Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients can grow superlinearly (in the space variable). These discontinuities are described by a piecewise local Lipschitz continuity and a piecewise monotone-type condition while the diffusion coefficient is assumed to be locally Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. Moreover, the superlinear nature of the coefficients is dictated by a suitable coercivity condition and a polynomial growth of the (local) Lipschitz constants of the coefficients. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate $1/2$, for a suitable range of values of $p$, is recovered for a tamed Euler scheme which is used for approximating these solutions. To the best of the authors' knowledge, these are the first existence, uniqueness and approximation results for this class of SDEs.

math.PR↗

arXiv2020

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and non-degenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best $L_p$-error rate obtained so far for approximation of scalar SDEs at the final time is $3/4$ in terms of the number of evaluations of the driving Brownian motion. In the present article we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the $L_p$-error rate $3/4$ can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the $L_p$-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the $L_p$-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables $(Y,Z)$ by providing explict lower bounds for the covariance $\text{Cov}(f(Y),g(Z))$ in case of piecewise Lipschitz continuous functions $f$ and $g$. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals $\int_0^1 f(W_t)\, dt$ of a Brownian motion $W$, which substantially extends known results for the case of $f$ being an indicator function.

math.PR↗

arXiv2020

Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven SDEs with smooth drift coefficient functions with at most polynomially growing derivatives

In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the the existence of a unique solution to the SDE) depend in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs which have smooth drift coefficient functions with at most polynomially growing derivatives whose solutions do not depend on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.

math.PR↗

arXiv2019

A strong order $3/4$ method for SDEs with discontinuous drift coefficient

In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently it has been shown in [Müller-Gronbach, T., and Yaroslavtseva, L., On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient, arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient the classical Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$. Up to now this was the best $L_p$-error rate available in the literature for equations of that type. In the present paper we construct a method based on finitely many evaluations of the driving Brownian motion that even achieves an $L_p$-error rate of at least $3/4$ for all $p\in [1,\infty)$ under additional piecewise smoothness assumptions on the coefficients. To obtain this result we prove in particular that a quasi-Milstein scheme achieves an $L_p$-error rate of at least $3/4$ in the case of coefficients that are both Lipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives, which is of interest in itself.

math.PR↗

arXiv2019

On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values

Recently in [M. Hairer, M. Hutzenthaler, and A. Jentzen, Ann. Probab. 43, 2 (2015), 468--527] and [A. Jentzen, T. Müller-Gronbach, and L. Yaroslavtseva, Commun. Math. Sci. 14, 6 (2016), 1477--1500] stochastic differential equations (SDEs) with smooth coefficient functions have been constructed which have an arbitrarily slowly converging modulus of continuity in the initial value. In these SDEs it is crucial that some of the first order partial derivatives of the drift coefficient functions grow at least exponentially and, in particular, quicker than any polynomial. However, in applications SDEs do typically have coefficient functions whose first order partial derivatives are polynomially bounded. In this article we study whether arbitrarily bad regularity phenomena in the initial value may also arise in the latter case and we partially answer this question in the negative. More precisely, we show that every additive noise driven SDE which admits a Lyapunov-type condition (which ensures the existence of a unique solution of the SDE) and which has a drift coefficient function whose first order partial derivatives grow at most polynomially is at least logarithmically Hölder continuous in the initial value.

math.PR↗

arXiv2018

On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient

Recently a lot of effort has been invested to analyze the $L_p$-error of the Euler-Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_p$-error rate of at least $1/(2p)-$ has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler-Maruyama scheme in fact achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$ as in the case of SDEs with Lipschitz coefficients.

math.NA↗

arXiv2017

Lower Error Bounds for Strong Approximation of Scalar SDEs with non-Lipschitzian Coefficients

We study pathwise approximation of scalar stochastic differential equations at a single time point or globally in time by means of methods that are based on finitely many observations of the driving Brownian motion. We prove lower error bounds in terms of the average number of evaluations of the driving Brownian motion that hold for every such method under rather mild assumptions on the coefficients of the equation. The underlying simple idea of our analysis is as follows: the lower error bounds known for equations with coefficients that have sufficient regularity globally in space should still apply in the case of coefficients that have this regularity in space only locally, in a small neighborhood of the initial value. Our results apply to a huge variety of equations with coefficients that are not globally Lipschitz continuous in space including Cox-Ingersoll-Ross processes, equations with superlinearly growing coefficients, and equations with discontinuous coefficients. In many of these cases the resulting lower error bounds even turn out to be sharp.

math.NA↗

arXiv2017

A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives

Recently, it has been shown in [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of autonomous stochastic differential equations (SDE) on the time interval $[0,T]$ with infinitely differentiable and bounded coefficients such that no strong approximation method based on evaluation of the driving Brownian motion at finitely many fixed times in $[0,T]$, e.g. on an equidistant grid, can converge in absolute mean to the solution at the final time with a polynomial rate in terms of the number of Brownian motion values that are used. In the literature on strong approximation of SDEs, polynomial error rate results are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition. This assumption is violated for the pathological SDEs from the above mentioned negative result. However, in the present article we construct an SDE with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can not be approximated with a polynomial rate, whatever method based on observations of the driving Brownian motion at finitely many fixed times is used. Most interestingly, it turns out that using a method that adjusts the number of evaluations of the driving Brownian motion to its actual path, the latter SDE can be approximated with rate 1 in terms of the average number of evaluations that are used. To the best of our knowledge, this is only the second example in the literature of an SDE for which there exist adaptive methods that perform superior to non-adaptive ones with respect to the convergence rate.

math.PR↗

arXiv2016

On hard quadrature problems for marginal distributions of SDEs with bounded smooth coefficients

In recent work of Hairer, Hutzenthaler and Jentzen, see [9], a stochastic differential equation (SDE) with infinitely often differentiable and bounded coefficients was constructed such that the Monte Carlo Euler method for approximation of the expected value of the first component of the solution at the final time converges but fails to achieve a mean square error of a polynomial rate. In the present paper we show that this type of bad performance for quadrature of SDEs with infinitely often differentiable and bounded coefficients is not a shortcoming of the Euler scheme in particular but can be observed in a worst case sense for every approximation method that is based on finitely many function values of the coefficients of the SDE. Even worse we show that for any sequence of Monte Carlo methods based on finitely many sequential evaluations of the coefficients and all their partial derivatives and for every arbitrarily slow convergence speed there exists a sequence of SDEs with infinitely often differentiable and bounded by one coefficients such that the first order derivatives of all diffusion coefficients are bounded by one as well and the first order derivatives of all drift coefficients are uniformly dominated by a single real-valued function and such that the corresponding sequence of mean absolute errors for approximation of the expected value of the first component of the solution at the final time can not converge to zero faster than the given speed.

math.NA↗

arXiv2015

On stochastic differential equations with arbitrary slow convergence rates for strong approximation

In 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 and globally bounded coefficients such that the Euler scheme converges to the solution in the strong sense but with no polynomial rate. Hairer et al.'s result naturally leads to the question whether this slow convergence phenomenon can be overcome by using a more sophisticated approximation method than the simple Euler scheme. In this article we answer this question to the negative. We prove that there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion converges in absolute mean to the solution with a polynomial rate. Even worse, we prove that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.

math.NA↗