Dimitra Antonopoulou (University of Athens, Greece) Finite Element Methods for a class of Stochastic PDEs from phase transitions

We shall consider a class of spdes from phase transitions with additive or multiplicative noise and we will present some continuous or discontinuous finite element schemes applied on their weak formulation. Results on the existence and regularity of stochastic solutions of the continuous problems, together with the noise approximation implemented, will be utilized in order to establish well-posedness of the discrete schemes and to estimate the numerical error.

Lubomir Banas (Universitat Bielefeld, Germany) Robust a posteriori estimates for the stochastic Cahn-Hilliard equation with singular noise

We discuss a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation with singular space-time noise. To deal with the low spatial regularity of the noise we consider a regularized problem with suitable approximation of the space-time white noise. The a posteriori bound for the regularized problem is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation and treating the respective problems separately. The estimate for the discretization of the original problem is then obtained by considering suitable probability subsets with high probability. The resulting a posteriori estimate is computable and robust with respect to the interfacial width parameter as well as the noise intensity. We illustrate the theoretical result by numerical simulations.

Christian Bayer (WIAS, Germany) A kernel regression approach to local stochastic volatility models

Perfect calibration of stochastic local volatility models can be achieved by the particle method due to Guyon and Henry-Labordère. Starting from a back-bone stochastic volatility model, a local volatility factor is computed on the fly to perfectly fit market prices. Mathematically, the local volatility factor is given as a conditional expectation, which is approximated by a local regression procedure. While this procedure is quite popular among practitioners, there are substantial gaps in the theoretical understanding. Indeed, even well-posedness of the resulting singular McKean-Vlasov system is not known. We develop a novel regularization approach based on the reproducing kernel Hilbert space technique (kernel ridge regression) and show that the regularized model is, in fact, well-posed. Furthermore, we prove propagation of chaos and provide error estimates for the numerical scheme. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models, and demonstrate excellent performance. Our results are also applicable to more general McKean--Vlasov equations. (Joint work with Denis Belomestny, Oleg Butkovsky, and John Schoenmakers.)

Denis Belomestny (Duisburg-Essen University, Germany) Forward Reverse Kernel Regression for the Schroedinger bridge problem

In this talk, we study the Schroedinger Bridge Problem (SBP), which is central to entropic optimal transport. I n the SBP, the reference Markov process (sometimes called the prior or baseline process) is the baseline stochastic dynamics against which one measures the relative entropy when solving for the new "bridge" process. The goal is to reweight or tilt this reference process into a new Markov process whose initial and terminal distributions match prescribed marginals, while remaining as close as possible in terms of Kullback-Leibler divergence to the original dynamics. For general reference processes, we propose a forward-reverse iterative reweighting procedure to approximate Schroedinger potentials in a nonparametric way. We use kernel-smoothing approximations combined with Picard iterations to preserve the positivity and contraction structure needed for the SBP iteration, leading to provably convergent algorithms. Furthermore, we provide convergence rates in the Hilbert metric for these potential estimates and prove their optimality. Finally, we discuss applications of our results to generative modeling. (joint with J. Schoenmakers)

Mireille Bossy (INRIA, France) Weak rough kernel comparison via PPDEs for integrated Volterra processes

Motivated by applications in physics (e.g., turbulence intermittency) and financial mathematics (e.g., rough volatility), this talk examines a family of integrated stochastic Volterra processes characterized by a small Hurst parameter H<1/2. We investigate the impact of kernel approximation on the integrated process by examining the resulting weak error. We quantify this error in terms of the L1 norm of the difference between the two kernels, as well as the L1 norm of the difference of the squares of these kernels. Our analysis is based on a path-dependent Feynman-Kac formula and the associated path-dependent partial differential equation (PPDE).

We apply this weak error estimation to further analyse some causal process propositions for intermittent turbulent transport (corresponding to the case where H goes to 0). This is a joint work with Kerlyns Martinez and Paul Maurer

Nawaf Bou-Rabee (Rutgers, USA) Introducing WALNUTS: A No-U-Turn Sampler with Locally Adaptive Step Size

Locally adapting parameters within Markov chain Monte Carlo methods while rigorously preserving reversibility has long posed a significant challenge. The success of the No-U-Turn Sampler (NUTS) largely stems from its clever local adaptation of the integration duration in Hamiltonian Monte Carlo via a geometric U-turn condition. However, posterior distributions frequently exhibit multi-scale geometries with extreme variations in scale, making it necessary to also adapt the leapfrog integrator’s step size locally and dynamically. This problem has remained open since the seminal introduction of NUTS by Hoffman and Gelman (2014).

To address this issue, we introduce WALNUTS (Within-Orbit Adaptive Step-Length No-U-Turn Sampler), a generalization of NUTS that finely adapts the leapfrog step size dynamically at each integration step within each orbit. Like NUTS, WALNUTS is rejection-free and employs biased progressive state selection to favor iterates furthest away from the orbit’s starting point. Through rigorous numerical experiments on challenging posterior distributions—including cases with pronounced funnel-shaped geometries—we demonstrate that WALNUTS substantially improves sampling efficiency, robustness, and effective exploration compared to standard NUTS.

This talk is based on joint work with Bob Carpenter (Flatiron), Tore Kleppe (Stavanger), Sifan Liu (Flatiron/Duke), and Milo Marsden (Stanford).

Charles-Edouard Bréhier (Université de Pau et des Pays de l'Adour, France) Analysis of stochastic optimization schemes using modified equations and weak error techniques

I will present the analysis of gradient descent and heavy ball algorithms containing stochastic perturbations. While at leading order the discrete time dynamics can be related to an ordinary differential equations, using modified or high-resolution equations it has been shown that higher order (weak) approximation is achieved. We are interested in proving the validity of this approximation in the large time regime, in order to derive complexity results on the considered stochastic algorithms. We will show such theoretical results and numerical experiments. This is based on joint works with Nassim En-Nebbazi and Marc Dambrine.

Dan Crisan (Imperial College London, UK) Particle filters for geophysical fluid dynamics using generative models

Particle filters are a class of numerical methods used for estimating the evolution of a signal process in the presence of uncertainty. They are particularly useful for nonlinear and non-Gaussian models, where traditional filtering methods like the Kalman filter fail. The key idea is to represent the posterior distribution of the system’s state using a set of weighted samples, or "particles," which are propagated and updated over time based on incoming observations. In this talk, I will present an application of the particle filtering methodology to a problem involving geophysical fluid dynamics. The dynamics model is calibrated using diffusion generative models. We generate synthetic data that statistically aligns with a given set of observations (in this case the increments of the numerical approximation of a solution of a partial differential equation). This allows us to efficiently implement a model reduction and assimilate data from a reference system state modelled by a highly resolved numerical solution of the rotating shallow water equation with 10,000 degrees of freedom into a stochastic system having two orders of magnitude less degrees of freedom. The new samples are incorporated into a particle filtering methodology augmented with tempering and jittering for dynamic state estimation, a method particularly suited for handling complex and multimodal distributions. This is joint work with Oana Lang (Babeș-Bolyai University) and Alexander Lobbe (Imperial College London) and is based on the paper:

Dan Crisan, Oana Lang, Alexander Lobbe, Bayesian inference for geophysical fluid dynamics using generative models, to appear in Philosophical Transactions of the Royal Society A, 2025.

Andreas Eberle (Universitat Bonn, Germany) Non-reversible lifts of reversible diffusion processes and relaxation times

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time.

For reversible diffusions on domains in Euclidean space, or, more generally, on a Riemannian manifold with boundary, non-reversible lifts are in particular given by the Hamiltonian flow on the tangent bundle, interspersed with random velocity refreshments, or perturbed by Ornstein-Uhlenbeck noise, and reflected at the boundary. In order to prove that for certain choices of parameters, these lifts achieve the optimal square-root reduction up to a constant factor, precise upper bounds on relaxation times are required. We demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincaré inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.

This is joint work with Francis Lörler (Bonn).

Monika Eisenmann (Lund University, Sweden) A fully discretized domain decomposition approach for semi-linear SPDEs

In recent years, SPDEs have become a well-studied field in mathematics. With their increasing popularity, it has become important to efficiently approximate their solutions. Therefore, our goal is to develop a modern time-stepping method for SPDEs. We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations. Our method is based on a non-iterative domain decomposition approach, which can help parallelize the code, leading to more efficient implementations. The domain decomposition is integrated through an operator splitting approach, where each operator acts on a specific part of the domain. More precisely, we combine the implicit Euler method with the Douglas-Rachford splitting scheme. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method. We provide a strong space-time convergence result for this fully discretized scheme and verify our theoretical findings through numerical experiments. This talk is based on a joint work with Eskil Hansen and Marvin Jans (both Lund University).

James Foster (University of Bath, UK) High order splitting methods for SDEs

In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain features of the Brownian motion. At the same time, the ODEs obtained from this path-based approach can be interpreted as stages in a splitting method, which neatly connects our work to the existing literature.

We give several examples to demonstrate the flexibility and convergence properties of our methodology. Most notably, for Underdamped Langevin Dynamics (ULD), we obtain a numerical method which exhibits third order convergence. When used as an MCMC algorithm, we empirically demonstrate its efficacy for performing Bayesian logistic regression against the No U-Turn Sampler (NUTS).

(joint work with Gonçalo dos Reis, Calum Strange and Andraž Jelinčič)

Máté Gerencsér (TU Wien, Austria) The Milstein scheme revisited

Emmanuel Gobet (CMAP, France) Walking forward and backward in Euler schemes and random number generators

Kristin Kirchner (TU Delft, Netherlands and KTH Royal Institute of Technology, Sweden) Numerical methods for the SPDE approach in space-time

Most environmental data sets contain measurements collected over space and time. It is the purpose of spatiotemporal statistical models to adequately describe the underlying uncertain spatially explicit phenomena evolving over time. In this I will present a new class of spatiotemporal statistical models which is based on stochastic partial differential equations (SPDEs) involving fractional powers of parabolic operators. In particular, I will discuss the efficient approximation of the covariance operators corresponding to the spatiotemporal latent Gaussian processes. The numerical methods will be based on space-time finite element discretizations of fractional parabolic operators, which are supported by a rigorous error analysis using inf-sup theory. Furthermore, I will address the motivation for employing this class of SPDEs in statistical applications and give an outlook on the computational benefits for statistical inference from spatiotemporal data.

Benedict Leimkuhler (University of Edinburgh, UK) SamAdams: an adaptive stepsize Langevin sampling algorithm inspired by the Adam optimizer

I will present a framework for adaptive-stepsize MCMC sampling based on time-rescaled Langevin dynamics, in which the stepsize variation is dynamically driven by an additional degree of freedom. The use of an auxiliary relaxation equation allows accumulation of a moving average of a local monitor function and while avoiding the need to modify the drift term in the physical system. Our algorithm is straightforward to implement and can be readily combined with any off-the-peg fixed-stepsize Langevin integrator. I will focus on a specific variant in which the stepsize is controlled by monitoring the norm of the log-posterior gradient, inspired by the Adam optimizer. As in Adam, the stepsize variation depends on the recent history of the gradient norm, which enhances stability and improves accuracy. I will discuss the application of our method to examples such as Neal’s funnel and a Bayesian neural network for classification of MNIST data. Joint work with Rene Lohmann (Edinburgh) and Peter Whalley (ETH).

Maud Lemercier (University of Oxford, UK) High order solvers for signature kernels

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernels of bounded variation paths, such as piecewise linear interpolations of time series data, are typically computed by solving a linear hyperbolic second-order PDE. However, this approach becomes considerably less practical for highly oscillatory inputs, due to significant time and memory complexities. To mitigate this issue, I will introduce a high order method which involves replacing the original PDE, which has rapidly varying coefficients, with a system of coupled equations with piecewise constant coefficients. These coefficients are derived from the first few terms of the log-signatures of the input paths and can be computed efficiently using existing Python libraries.

Chengcheng Ling (Augsburg University, Germany) The Milstein scheme for singular SDEs with Hölder continuous drift

We study the rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.

Gabriel Lord (Radboud University, Netherlands) A numerical method for domain preservation

Many stochastic (partial) differential equations that arise in mathematical biology preserve positivity (eg concentrations should be positive) or solutions should lie in a certain domain (eg [0,1] for stochastic gating variables in neurosciene). The question then arises how to preserve these domains in a numerical simulation. We develop a numerical method based on exponential integrators that naturally preserve such domains and discuss convergence and efficiency. This work is joint with Utku Erdogan.

Pierre Monmarché (Sorbonne, France) Non-asymptotic error bounds for unadjusted HMC and Langevin for mean-field models

We consider a family of MCMC samplers ranging from Hamiltonian Monte Carlo to Langevin diffusion, for which we establish non-asymptotic convergence bounds in relative entropy. The bound is sharp in terms of the step-size and the log-Sobolev constant of the target measure. In particular this result is suitable to deal with mean-field models, namely particle systems used to approximate non-linear evolutions or arising for instance in models from statistical physics or machine learning algorithms. We will briefly mention the case of metastable mean-field models for which the asymptotic convergence rate to equilibrium is very small but, still, a quick convergence to a local non-linear stationary state occurs and remains true for times which are exponentially large with the number of particles, which is the interesting behavior in practice.

Cornelis Oosterlee (Utrecht University, Netherlands) On a Fourier cosine expansion method with higher order Taylor schemes for fully coupled FBSDEs

A higher-order numerical method is presented for scalar valued, coupled forward-backward stochastic differential equations. Unlike most classical references, the forward component is not only discretized by an Euler-Maruyama approximation but also by higher-order Taylor schemes. This includes the famous Milstein scheme, providing an improved strong convergence rate of order 1; and the simplified order 2.0 weak Taylor scheme exhibiting weak convergence rate of order 2. In order to have a fully-implementable scheme in case of these higher-order Taylor approximations, which involve the derivatives of the decoupling fields, we use the COS method built on Fourier cosine expansions to approximate the conditional expectations arising from the numerical approximation of the backward component. Even though higher-order numerical approximations for the backward equation are deeply studied in the literature, to the best of our understanding, the present numerical scheme is the first which achieves strong convergence of order 1 for the whole coupled system, including the forward equation, which is often the main interest in applications such as stochastic control. Numerical experiments demonstrate the proclaimed higher-order convergence, both in case of strong and weak convergence rates, for various equations ranging from decoupled to the fully-coupled settings. This is joint work with Balint Negyesi.

Michela Ottobre (Heriot Watt, UK) Interacting particle systems, McKean-Vlasov PDEs and S(P)DEs with additive noise

When dealing with systems which are made of a large number of particles, one is often interested in the collective behaviour of the system rather than in a detailed description. Established approaches in statistical mechanics and kinetic theory allow one to study the limit as the number of particles N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Of course one of the issues here is which properties of the initial particle systems are preserved upon passing to the limit – and this is something we will touch upon. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is further more multiplicative and has gradient structure. One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems. We will moreover discuss applications of the problem of sampling from the invariant distribution of SPDEs with additive noise.

Andreas Prohl (Tuebingen University, Germany) Numerical Methods for Optimal Control Problems with SPDEs

I compare different numerical methods for solving stochastic linear quadratic (SLQ) optimal control problems governed by stochastic partial differential equations (SPDEs). The constructions of numerical methods rest on the open-loop and closed-loop approaches. The open-loop approach involves the complexities of coupled forward-backward SPDEs and employs a gradient descent framework. In contrast, the closed-loop approach applies a feedback strategy, focusing on Riccati equation for spatio-temporal discretization. Both approaches lead to implementable fully discrete SLQ problems, resulting methods are settled by rigorous convergence analyses (spatio-temporal rates of convergence), and their efficiencies are compared by computational studies. The talk is based on recent joint works with Y. Wang (U Chongqing).

Goncalo dos Reis (University of Edinburgh, UK) Simulation of mean-field SDEs: some recent results

We review two results in the simulation for SDE of McKean-Vlasov type (MV-SDE). The first block of results addresses simulation of MV-SDEs having super-linear growth in the spatial and the interaction component in the drift, and non-constant Lipschitz diffusion coefficient. The 2nd block is far more curious. It addresses the study the weak convergence behaviour of the Leimkuhler–Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean–Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate 3/2) than the standard Euler method (of weak order 1). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests.

This presentation is based on the joint work [1], [2].

References: [1] Chen, X., Dos Reis, G., Stockinger, W. and Wilde, Z., 2025. Improved weak convergence for the long-time simulation of mean-field Langevin equations. Electronic Journal of Probability, 30, pp.1-81.
[2] X. Chen, G. dos Reis. "Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction" IMA Journal of Numerical Analysis, 44, no. 2 (2024), 751-796.

Denis Talay (Inria Saclay Ile de France, France) A novel methodology to test infinite expectations

This is a joint work with Hector Olivero (Universidad de Valparaiso). Motivated by the simulation of stochastic particle systems with singular McKean-Vlasov interaction kernels we address the problem of detecting whether a sampled probability distribution has infinite expectation. As stated, the detection problem is ill-posed. We thus propose and analyze an asymptotic hypothesis test for random variables which belong to an unknown domain of attraction of a stable law. The null hypothesis : `The random variable interest is in the domain of attraction of the Normal law' and the alternative hypothesis is `The random variable of interest is in the domain of attraction of a stable law with index smaller than 2'. Our key observation is that the random variable cannot have a finite second moment when the null hypothesis is rejected. Surprisingly, we find it useful to derive our test from the statistics of stochastic processes: More precisely, from tests aimed to determine whether a semimartingale has jumps by observing one single path at discrete times. We will discuss the choice of crucial parameters involved in the test and illustrate our theoretical results with numerical experiments.

Irene Tubikanec (Johannes Kepler University, Austria) Network inference via approximate Bayesian computation. Illustration on a stochastic multi-population neural mass model

In this talk, we propose an adapted sequential Monte Carlo approximate Bayesian computation (SMC-ABC) algorithm for network inference in coupled stochastic differential equations (SDEs) used for multivariate time series modeling. Our approach is motivated by neuroscience, specifically the challenge of estimating brain connectivity before and during epileptic seizures. To this end, we make four key contributions. First, we introduce a 6N-dimensional SDE to model the activity of N coupled neuronal populations, extending the (single-population) stochastic Jansen and Rit neural mass model used to describe human electroencephalography (EEG) rhythms, particularly epileptic activity. Second, we construct a reliable and efficient numerical splitting scheme for the model simulation. Third, we apply the proposed adapted SMC-ABC algorithm to the neural mass model and validate it on different types of simulated data. Compared to standard SMC-ABC, our approach significantly reduces computational cost by requiring fewer model simulations to reach the desired posterior region, thanks to the inclusion of binary parameters describing the presence or absence of coupling directions. Finally, we apply our method to real multi-channel EEG data, uncovering potential similarities in patients' brain activities across different epileptic seizures, as well as differences between pre-seizure and seizure periods.

Gilles Vilmart (Université de Genève, Switzerland) High-order sampling of the invariant distribution of ergodic stochastic dynamics: preconditioning and postprocessing

Preconditioning techniques for ergodic stochastic dynamics are attractive to enhance the convergence to the invariant measure especially in high dimension, to tackle the issues of stiffness and multi-modal target distributions. Achieving high order of accuracy is however challenging in the context. In this talk, we present an integrator for overdamped Langevin dynamics with position dependent diffusion tensor, with a postprocessor that is second order accurate for sampling the invariant measure, while requiring only one force evaluation per timestep. Analysis of the sampling bias is performed using the algebraic framework of exotic aromatic Butcher-series. This talk is based on joint work with Eugen Bronasco (Univ. Geneva), Benedict Leimkuhler, and Dominic Phillips (Univ. Edinburgh). Preprints and papers available at: https://www.unige.ch/~vilmart

Yue Wu (University of Strathclyde, UK) The randomised quadrature and randomised numerical schemes

In this talk, I will explain how stratified Monte Carlo simulation can be used to develop randomised quadrature methods for time integration (1D case) and integration over a 2D domain (2D case). Building on these foundations, I will present randomised numerical schemes applicable to a broad class of deterministic differential equations with irregular coefficients. The initial schemes considered are randomised Euler and randomised Runge-Kunta for Caratheodory ODEs. Our motivation for studying Caratheodory type initial value problems stems from the fact that certain rough differential equations that are driven by an additive noise can be transformed into a problem of this form. It is well-known that there is lack of convergence for deterministic algorithms in this case, and our proposed methods in [1] showed that even with very mild conditions, the order of convergence can be at least half with respect to the Lp norm. We then extend it to a stochastic setting by considering the draft-randomised Milstein method [2]. We then investigated the applications of two randomised quadrature formulas (2D case) to the finite element method (FEM) for elliptic boundary value problems with irregular coefficient functions in [3]. In general, the entries of this matrix-vector system of FEM are not known explicitly but need to be approximated by quadrature rules. If the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. Our analysis showed that the convergence rate of our proposed quadrature can be at least half-2/p even if the force term is only Lp integrable. Recently, we re-examined the randomised quadrature rules (1D) using stochastic sewing lemma [4] and constructed an almost-optimal convergence when approximating the solutions of additive SDEs with irregular drifts [5].

[1] Kruse, R. and Wu, Y., 2017. Error analysis of randomised Runge–Kutta methods for differential equations with time-irregular coefficients. Computational Methods in Applied Mathematics, 17(3), pp.479-498.
[2] Kruse, R. and Wu, Y., 2019. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete and Continuous Dynamical Systems - Series B, 24(8), pp.3475-3502.
[3] Kruse, R., Polydorides, N. and Wu, Y., 2019. Application of Randomised Quadrature Formulas to the Finite Element Method for Elliptic Equations. arXiv preprint arXiv:1908.08901.
[4] Le, K.: A stochastic sewing lemma and applications, Electronic Journal of Probability, 25(38), 1-55, (2020).
[5] Bao, J. and Wu, Y., 2025, Randomised Euler-Maruyama method for SDEs with Hoelder continuous drift coefficient. arXiv preprint, arXiv:2501.15527.

Larisa Yaroslavtseva (University of Graz, Austria) On sharp lower bounds for strong approximation of SDEs with Hölder continuous drift coefficient

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 α-Hölder continuous with α ∈ (0, 1]. Recently, it has been shown in [1] that for such SDEs the equidistant Euler approximation achieves an L_p-error rate of at least (1+α)/2, up to an arbitrary small ϵ >0, in terms of the number of evaluations of the driving Brownian motion. In this talk we present a matching lower error bound. More precisely, we show that the L_p-error rate (1+α)/2 can not be improved in general by no numerical method based on finitely many evaluations of the driving Brownian motion at fixed time points. For the proof of this result, we choose the drift coefficient to be the Weierstrass function and we employ the coupling of noise technique recently introduced in [2].

The talk is based on a joint work with Simon Ellinger and Thomas Müller-Gronbach (both from the University of Passau).

[1] Butkovsky, O., Dareiotis, K., and Gerencs\'er, M. Approximation of SDEs: a stochastic sewing appproach. Probab. Theory Related Fields 181, 4 (2021), 975–1034
[2] Müller-Gronbach, T., and Yaroslavtseva, L. Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise. Ann. Appl. Probab. 33 (2023), 902–935.