Projects

The overall aim of this proposed research is to obtain new theoretical understanding of how randomized numerical methods (RNMs) can improve simulations of stochastic models where traditional methods do not provide satisfactory results, and how they can be extended to apply in the general area of evolving systems and optimal control.The RNMs were initially jointly proposed by the lead applicant to eradicate the dilemma between “slow-and-accurate” or “fast-and-crude” for approximating ordinary differential equations with irregular coefficients. RNMs combine two established approaches to equation-solving: numerical methods with error propagation and probabilistic representations of equations using simulation techniques like Monte Carlo. Consequently, RNMs inherit the strengths of these two approaches, resulting in improved numerical efficiency compared to the conventional numerical methods they are based on. This advantage was later extended to stochastic scenarios, enabling the efficient approximation of a wide range of nonlinear stochastic differential equations (SDEs). While existing numerical methods for nonlinear SDEs often entail high computational costs due to implicitness or the need for fine step sizes, empirical experiments have revealed that their randomized counterparts can exhibit super-efficiency. This includes significant diffusion models like the Cox-Ingersoll-Ross model, which lacks analytically explicit solutions. Uncovering the rationale behind the outstanding performance holds significant importance, as it has the potential to illuminate broader applications for RNMs.In reconsidering the fundamentals of such computational designs, we will address three principal objectives:To generalise the existing framework of randomised numerical methods to provide effective and reliable simulations for stochastic models that are widely used in finance, ecology, physics, and engineering. This will involve creating novel and innovative numerical technologies for nonlinear stochastic differential equations and time-changed nonlinear stochastic differential equations, with or without jump diffusions.To provide a rigorous error analysis by quantifying the convergence errors under suitable topologies as well as by evaluating the performance through various numerical experiments.To increase the visibility of randomised numerical schemes and to accelerate their adoption by the scientific community and by non-scholars such as industrial users. We will develop and standardise a publicly-available Python library for randomised numerical schemes with access to various types of differential equations and applications.In summary, this project will stimulate methodological developments in the design of effective randomised numerical methods and create knowledge that promotes transformative and cross-disciplinary research.

• University of Strathclyde

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Optimization problems accompany us all the time in our every-day life. For example supermarkets guarantee the supply by optimizing the route of transportation of their goods, electricity providers optimize the supply with electricity, and highways are built in a way such that cars produce as little noise as possible. For this, discrete, stochastic, which means influenced by randomness, and continuous mathematical models are used, which need to be analyzed. In order to solve such problems, it is often essential to have a multi-perspective view and combine the knowledge of several mathematical sub-disciplines to create synergies. In crossing these borders lies a great innovative potential. It is the aim of the doc.funds doctoral school and its nine professors from the Departments of Mathematics and Statistics at the University of Klagenfurt to provide PhD students with the mathematical knowledge necessary for understanding and solving challenging mathematical questions, coming from optimization problems in every-day life.

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Recent years have seen a tremendous surge in the availability of socioeconomic data characterized by vast complexity and high dimensionality. However, prevalent methods employed to inform practitioners and policy makers are still focused on small to medium-scale datasets. Consequently, crucial transmission channels are easily overlooked and the corresponding inference often suffers from omitted variable bias. This calls for novel methods which enable researchers to fully exploit the ever increasing amount of data. In this project, we aim to investigate how the largely separate research streams of Bayesian econometrics, statistical model checking, and machine learning can be combined and integrated to create innovative and powerful tools for the analysis of big data in economics and other social sciences. Thereby, we pay special attention to properly incorporating relevant sources of uncertainty. Albeit crucial for thorough empirical analyses, this aspect is often overlooked in traditional machine learning techniques which have mainly been centered on producing point forecasts for key quantities of interest only. In contrast, Bayesian statistics and econometrics are based on designing algorithms to carry out exact posterior inference which in turn allows for density forecasts. Our contributions are twofold: From a methodological perspective, we develop cutting-edge methods that enable fully probabilistic inference of dynamic models in vast dimensions. In terms of empirical advances, we apply these methods to highly complex datasets that comprise situations where either the number of observations, the number of potential time series and/or the number of variables included is large. More specifically, empirical applications center on four topical issues in the realm of sustainable development and socioeconomic policy to answer questions such as: How do market and economic uncertainty affect income inequality? What are the relationships between greenhouse gas emissions and macroeconomic indicators? Which role do tweets play in the evolution of the prices of crypto-currencies? Which policy measures are most effective to foster sustainable urban mobility patterns? In these applications, we focus on probabilistic forecasting using real-time data to perform model validation in an efficient way. Moreover, we address structural inference. As policy makers are typically interested in evaluating their policies quantitatively, robust econometric tools are crucial for counterfactual simulations. In light of the increasing complexity of the economy, however, large information sets need to be exploited to appropriately recover the underlying causal structures and provide a rich picture of potential transmission channels of policy interventions. The team constitutes a genuinely collaborative partnership of five young high-potential researchers composed of statisticians, machine learning experts, macro- and regional economists as well as social and computer scientists.

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