arXiv subjects

Simon Ellinger

Publications and source records attributed to Simon Ellinger.

5 recordsLinked to original sources

arXiv2025

The local coupling of noise technique and its application to lower error bounds for strong approximation of SDEs with irregular coefficients

In recent years, interest in approximation methods for stochastic differential equations (SDEs) with non-Lipschitz continuous coefficients has increased. We show lower bounds for the $L^p$-error of such methods in the case of approximation at a single point in time or globally in time. On the one hand, we show that for a large class of piecewise Lipschitz continuous drifts and non-additive diffusions the best possible $L^p$-error rate for final time approximation that can be achieved by any method based on finitely many evaluations of the driving Brownian motion is at most $3/4$, which was previously known only for additive diffusions. Moreover, we show that the best $L^p$-error rate for global approximation that can be achieved by any method based on finitely many evaluations of the driving Brownian motion is at most $1/2$ when the drift is locally bounded and the diffusion is locally Lipschitz continuous. For the derivation of the lower bounds we introduce a new method of proof: the local coupling of noise technique. Using this technique when approximating a solution $X$ of the SDE at the final time, a lower bound for the $L^p$-error of any approximation method based on evaluations of the driving Brownian motion at the points $t_1 < \dots < t_n$ can be determined by the $L^p$-distances of solutions of the same SDE on $[t_{i-1}, t_i]$ with initial values $X_{t_{i-1}}$ and driving Brownian motions that are coupled at $t_{i-1}, t_i$ and independent, conditioned on the values of the Brownian motion at $t_{i-1}, t_i$.

math.PR↗

arXiv2025

Regularity properties of densities of SDEs using the Fourier analytic approach

We show regularity properties of local densities of solutions of stochastic differential equations (SDEs) with the Fourier analytic approach. With this simple method, statements that were previously derived with approaches using Malliavin calculus or difference operators can be recovered and extended to include regularity properties with respect to the time variable. For example, we derive the H\"older continuity and joint continuity of local densities in the case of drift coefficients that are locally piecewise H\"older continuous. To this end, we derive fairly general bounds for the Fourier transform of the local density of a solution of the SDE when the drift is locally bounded and the diffusion is locally sufficiently regular.

math.PR↗

arXiv2025

Sharp lower error bounds for strong approximation of SDEs with a drift coefficient of H\"older or Sobolev regularity using a Weierstra{\ss} scale

We study strong approximation of solutions of SDEs with bounded $\alpha$-H\"older 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 $(1+\alpha)/2$, up to an arbitrary small $\varepsilon$, for all $p\geq 1$ and $\alpha\in (0,1]$, in terms of the number of evaluations of the driving Brownian motion $W$. In this article, we prove a matching lower error bound for $\alpha\in (0,1)$. More precisely, we show that for every $\alpha\in (0,1)$, the $L^p$-error rate $(1+\alpha)/2$ 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 $\alpha=1$. Even stronger, an $L^p$-error rate better than $(1+\alpha )/2$ 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 $\alpha\in (0,1)$. We prove that for every $\alpha\in (0,1)$, the $L^p$-error rate $(1+ \alpha )/2$ 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 $\alpha\in (1/2,1)$, $p=2$ 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↗

arXiv2024

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all $p \in [1, \infty)$ a transformed Milstein-type scheme reaches an $L^p$-error rate of at least $3 / 4$ when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate $3 / 4$ is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound of Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

math.PR↗