We study strong approximation of the solution of a scalar stochastic differential equation (SDE) \begin{equation}\label{sde0} \begin{aligned} dX_t & = \mu(X_t) \, dt + dW_t, \quad t\in [0,1], X_0 & = x_0 \end{aligned} \end{equation} at the final time point $1$ in the case that the drift coefficient $\mu$ is $\alpha$-H\"older continuous with $\alpha\in(0, 1]$. Recently, it has been shown in [1] that for such SDEs the equidistant Euler approximation achieves an $L^p$-error rate of at least $(1+\alpha)/2$, up to an arbitrary small $\varepsilon$, in terms of the number of evaluations of the driving Brownian motion $W$. In this talk we present a matching lower error bound. More precisely, we show that the $L^p$-error rate $(1+\alpha)/2$ can not be improved in general by no numerical method based on finitely many evaluations of $W$ at fixed time points. For the proof of this result we choose $\mu$ to be the Weierstrass function and we employ the coupling of noise technique recently introduced in [2]. [1] O. Butkovsky, K. Dareiotis, and M. Gerencser, Approximation of SDEs: a stochastic sewing appproach, Probab. Theory Related Fields 181, 4 (2021), 975--1034. [2] T. Müller-Gronbach and L. Yaroslavtseva, Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise. Ann. Appl. Probab. 33 (2023), 902--935.

We consider adaptive increasingly rare Markov chain Monte Carlo (AIR MCMC), which is an adaptive MCMC method, where the adaptation concerning the "past" happens less and less frequently over time. We are interested in the convergence behaviour of renormalised Monte Carlo sums and show limit results which hold under a Wasserstein contraction assumption. Our results hold in an almost sure setting and we obtain rates which are close to those in the law of the iterated logarithm. This talk is based on joint work with K. {{\L}atuszy\'{n}ski}, G. Roberts and D. Rudolf.

The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this work, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov’s transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method’s effectiveness, including the extension to time-dependent thresholds, comparisons with existing methods through numerical examples.

Let K be a smooth convex body, that is, a compact convex set with nonempty interior and three times contin- uously differentiable boundary. Assume that everywhere on the boundary the Gaussian curvature is positive. We can approximate K by the convex hull Kn of n random independent identically distributed points sampled from its boundary ∂K. It is known that the Hausdorff distance between K and the approximation Kn tends to zero as n tends to infinity and the speed of this convergence was determined by Glasauer and Schneider [1]. In case the points are distributed according to the optimal density (which is given in terms of the curvature), we prove that the rescaled Hausdorff distance between K and Kn tends to a Gumbel distributed random variable. The proof is based on an asymptotic relation to the covering properties of random geodesic balls on ∂K and relies on a theorem due to Janson [2]. References: [1] S. Glasauer, R. Schneider: Asymptotic approximation of smooth convex bodies by polytopes, Forum. Math., 8(3), 363–377, 1996. [2] S. Janson: Maximal spacings in several dimensions, Ann. Probab., 15(1), 274–280, 1987.

Modeling chemical reaction systems using sets of coupled ordinary differential equations (ODEs), which are then solved numerically, can be effective for small-scale reaction networks. However, as systems become larger, more complex, or exhibit combinatorial behavior, this deterministic approach often fails to capture important features such as concurrent reaction events, strong nonlinearities, or possible alternative reaction pathways—common characteristics of such systems. Stochastic simulation methods, like Gillespie-type algorithms [1], offer a more suitable alternative by accounting for randomness in reaction dynamics. Yet, these methods still require prior knowledge of the complete reaction network. Rule-based modeling [2] addresses this limitation by enabling network-free or on-the-fly stochastic simulations [3]. Instead of relying on a predefined reaction network, this approach operates on a set of reaction rules, each with associated rates. As the simulation progresses, the reaction network is generated dynamically based on the application of these rules. In the case of chemical reactive systems, molecules are modeled as graphs, and reactions are implemented as graph transformation rules [4]. This framework allows for rate calculations that can depend on the structural properties of molecules—enabled through customizable callback functions. Additionally, these rate estimates can be further refined using on- the-fly calculations of physicochemical properties, such as energies of formation. This modeling approach was applied to study the recently introduced glyoxylose chemical space [5], a system structurally analogous to the prebiotically significant formose reaction. The results of this investigation will be presented in this talk. Overall, this foundational modeling framework holds promise for future studies aimed at estimating kinetic parameters in experimental chemical systems.

This talk addresses a broad class of stochastic parabolic partial differential equations (PDEs) involving singular potentials, where the potential terms can be highly irregular. We consider several types of such equations, motivated by a range of applications from finance, structural mechanics, fluid dynamics and biology, where modeling uncertainty through stochastic effects is essential. To rigorously study these problems, we employ the Wick product from white noise analysis, a regularization technique that enables meaningful multiplication of generalized stochastic processes. The analysis is carried out within the framework of chaos expansions and combines tools from white noise analysis with the concept of very weak solutions in the theory of PDEs. For each class of equations considered, we formulate a notion of a stochastic very weak solution and establish results on existence and uniqueness. Moreover, we demonstrate that when the potential and input data are sufficiently regular, the stochastic very weak solution agrees with the classical stochastic weak solution. The talk is based on the joint work with Ljubica Oparnica and Sneˇzana Gordi´c.

In this talk, we will discuss the question of how far the particles of a supercritical branching random walk travel from their starting position, which is a quantity known as the maximal displacement of the branching random walk. This question has been extensively studied on the real line, where it is known that the maximal distance to the starting vertex grows linearly in time, and even the second order correction term has been determined to be of logarithmic order. However, on other state spaces, our understanding of the maximal distance travelled by the particles of a supercritical branching random walk is far less clear than on the real line. In my presentation, I will discuss the maximal displacement of supercritical branching random walks on free products of finite groups, where we are able to show that the maximal distance travelled also grows linearly in time almost surely. We will see how the maximal displacement relates to large deviation estimates of the underlying step distribution, and how we can use such estimates to prove the linear growth of the maximal displacement.

Nonlinear time series models with exogenous regressors play a pivotal role in econometrics, queuing theory, and machine learning, offering robust frameworks for modeling and prediction. Despite their widespread utility, the statistical analysis of such models remains challenging, particularly when dealing with non-i.i.d. exogenous covariate processes. Fundamental results, including the law of large numbers, the central limit theorem, and concentration inequalities, have been established for weakly dependent variables. We demonstrate the transfer of mixing properties from exogenous regressors to the response variable through coupling arguments, making applicable these results, thus providing a powerful tool for analyzing nonlinear autoregressions with exogenous inputs. Furthermore, we investigate Markov chains in random environments under suitable versions of drift and minorization conditions, including non-stationary environments with favorable mixing properties. This framework is then applied to single-server queuing models, opening door to the statistical analysis of waiting times.

We study and solve the worst-case optimal portfolio problem of an investor with logarithmic preferences facing the possibility of a market crash. Our setting takes place in a Levy-market and we assume stochastic market coefficients. To tackle this problem, we enhance the martingale approach developed in [1]. A utility crash-exposure transformation into a backward stochastic differential equation (BSDE) setting allows us to characterize the optimal indifference strategies. Further, we deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. To numerically compute the strategies, we solve the corresponding (non-Lipschitz) BSDEs through their associated PDEs and need to analyze continuity and boundedness properties of CIR forward processes. We demonstrate our approach for Heston’s stochastic volatility model, Bates’ stochastic volatility model including jumps, and Kim-Omberg’s model for a stochastic excess return. References: [1] F. T. Seifried: Optimal investment for worst-case crash scenarios: A martingale approach, Mathematics of Operations Research, 35 (2010), 559–579.