We study Poisson's equation in the context of general state space Markov chains. For chains satisfying a contraction assumption w.r.t. a Wasserstein distance we show that a solution exists for forcing functions which are Lipschitz. If the solution of Poisson's equation is sufficiently regular, then a martingale decomposition technique can be employed to investigate the convergence of empirical averages corresponding to the Markov chain. Examples from the Markov chain Monte Carlo (MCMC) literature which satisfy the contraction assumption are provided. Additionally, we show statements concerning the almost sure convergence of the corresponding MCMC estimators for numerical integration.

Continuous Integration (CI) has become an essential practice in modern software development, enabling rapid feedback through automated building and testing of software systems. However, projects grow in size and complexity. As a consequence, such CI pipelines increasingly suffer from high execution costs, dependency-related failures, and build breakages that require substantial developer effort to diagnose and repair.In this talk, I will present my research agenda on intelligent build engineering that combines prediction, validation, and automated repair techniques to improve the efficiency and reliability of software builds. First, I will discuss approaches for anticipating build outcomes and identifying skippable CI commits, leveraging both changes to source code and to build code, along with complexity measures, to enable more efficient CI utilization. Next, I will introduce my approaches for validating dependencies in build configurations and automatically detecting and repairing dependency conflicts, including approaches that leverage large language models to modify source code when configuration-level fixes are insufficient.Finally, I will explore the emerging role of AI agents in build maintenance. This includes evaluating whether large language models can replace traditional build log analyzers, as well as presenting agent-based approaches for diagnosing dependency-related failures and repairing such build breakages automatically. In summary, my research directions investigate how predictive analytics, dependency management, large language models, and autonomous agents may enable future CI systems that are faster, more resilient, and increasingly self-healing.

The first-passage time 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 talk, we discuss the extension of an existing exact simulation method, originally developed for constant thresholds, to the case of time-dependent thresholds. Our proposed approach utilises 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-discretisation errors, and directly simulates the FPT. We present results demonstrating the method’s effectiveness, including the extension to time-dependent thresholds, and comparisons with existing methods through numerical examples.

In filtering theory one seeks to analyze the conditional expectation of a signal process, given an observation process. In some cases it is possible to prove the existence of the conditional density, by solving a related stochastic partial differential equation (SPDEs). However, if both, signal and observation, are given by SDEs driven by discontinuous noise, the analysis of such SPDEs can be challenging. In this talk, I will present methods to prove existence of a conditional density to such systems, as well as give conditions to ensure a certain spatial (Sobolev) regularity of the conditional density. If time permits, I will give an outlook on future work.This is based on joint work with Alexander Davie and Istvan Gyongy.