Thorsten Schmidt

Many rare diseases offer limited established treatment options, leading patients to switch therapies when new medications emerge. To analyze the impact of such treatment switches within the low sample size limitations of rare disease trials, it is important to use all available data sources. This, however, is complicated when the use of measurement instruments changes during the observation period, for example when instruments are adapted to specific age ranges. The resulting disjoint longitudinal data trajectories complicate the application of traditional modeling approaches like mixed‐effects regression. We tackle this by mapping observations of each instrument to an aligned low‐dimensional temporal trajectory, enabling longitudinal modeling across instruments. Specifically, we employ a set of variational autoencoder architectures to embed item values into a shared latent space for each time point. Temporal disease dynamics and treatment switch effects are then captured through a mixed‐effects regression model applied to latent representations. To enable statistical inference, we present a novel statistical testing approach that accounts for the joint parameter estimation of mixed‐effects regression and variational autoencoders. The methodology is applied to quantify the impact of treatment switches for patients with spinal muscular atrophy. Here, our approach aligns motor performance items from different measurement instruments for mixed‐effects regression and maps estimated effects back to the observed item level to quantify the treatment switch effect. Our approach allows for model selection as well as for assessing effects of treatment switching. The results highlight the potential of modeling in joint latent representations for addressing small data challenges.

Medium- and long-term insurance products must participate in equity market returns to remain competitive. To eliminate downside risk, guarantees should be included, which naturally gives rise to the problem of valuing contracts in a unified insurance-finance framework. We study a general setup that allows for joint modelling of financial markets, mortality, and policy holder behaviour. Within this framework, we propose a general affine approach exploiting the tractability of affine processes to obtain explicit valuation formulas for variable annuities and related contracts. The model permits flexible dependence between mortality and equity dynamics. Moreover, surrender intensities are modelled as functions of the driving affine process, thereby introducing endogenous market dependence into lapse behaviour – a feature rarely treated analytically in the literature. The resulting framework combines analytical tractability with sufficient flexibility to capture key features of long-term insurance products.

To make medium- and long-term insurance products attractive, it is essential to enable participation in stock market returns. However, to eliminate downside risk, guarantees must be included, which naturally leads to the challenge of valuing such contracts within a unified insurance-finance framework. We develop a general setup that allows for the joint modelling of financial markets, mortality and policyholder behaviour. Within this framework, we propose a general affine approach and derive explicit valuation formulas for variable annuities and related contracts, while retaining computational tractability due to the affine structure. The model permits flexible dependence between mortality and equity dynamics, as supported by empirical evidence from the COVID-19 pandemic. Moreover, surrender intensities are modelled as functions of the underlying affine process, thereby introducing market dependence into lapse behaviour. The resulting framework combines analytical tractability with sufficient flexibility to capture key features of long-term insurance products.

We study an extension of the Cox-Ingersoll-Ross (CIR) process that incorporates jumps at deterministic dates, referred to as stochastic discontinuities. Our main motivation stems from short-rate modelling in the context of overnight rates, which often exhibit jumps at predetermined dates corresponding to central bank meetings. We provide a formal definition of a CIR process with stochastic discontinuities, where the jump sizes depend on the pre-jump state, thereby allowing for both upwarrd and downward movements as well as potential autocorrelation among jumps. Under mild assumptions, we establish existence of such a process and identify sufficient and necessary conditions under which the process inherits the affine property of its continuous counterpart. We illustrate our results with practical examples that generate both upward and downward jumps while preserving the affine property and non-negativity. In particular, we show that a stochastically discontinuous CIR process can be constructed by applying a determinisitic cadlag time-change of a classical CIR process. Finally, we further enrich the affine framework by characterizing conditions that ensure infinite divisibility of the extended CIR process.

We introduce a new mean-field game framework to analyze the impact of carbon pricing in a multi-sector economy with defaultable firms. Each sector produces a homogeneous good, with its price endogenously determined through market clearing. Firms act as price takers and maximize profits by choosing an optimal allocation of inputs-including labor, emissions, and intermediate goods from other sectors-while interacting through the endogenous sectoral price. Firms also choose their default timing to maximize shareholder value. Formally, we model the economy as an optimal stopping mean-field game within each sector. The resulting system of coupled mean-field games admits a linear programming formulation that characterizes Nash equilibria in terms of population measure flows. We prove the existence of a linear programming Nash equilibrium and establish uniqueness of the associated price system. Numerical illustrations are presented for firms with constant elasticity of substitution (CES) production functions. In a stylized single-sector economy, carbon price shocks induce substitution between emissions and labor. In a three-sector economy, the manufacturing sector faces consumer demand and requires inputs from a brown sector, which can be increasingly replaced by green-sector goods as carbon prices rise. These experiments reveal that carbon price shocks can generate substantial spillover effects along the value chain, underscoring the importance of sectoral interdependencies in shaping effective decarbonization pathways.

In this paper we study the pricing and hedging of nonreplicable contingent claims, such as long-term insurance contracts like variable annuities. Our approach is based on the benchmark-neutral pricing framework of Platen (2024), which differs from the classical benchmark approach by using the stock growth optimal portfolio as the numéraire. In typical settings, this choice leads to an equivalent martingale measure, the benchmark-neutral measure. The resulting prices can be significantly lower than the respective risk-neutral ones, making this approach attractive for long-term risk-management. We derive the associated risk-minimizing hedging strategy under the assumption that the contingent claim possesses a martingale decomposition. For a set of nonreplicable contingent claims, these strategies allow monitoring the working capital required to generate their payoffs and enable an assessment of the resulting diversification effects. Furthermore, an algorithmic refinancing strategy is proposed that allows modeling the working capital. Finally, insurance-finance arbitrages of the first kind are introduced and it is demonstrated that benchmark-neutral pricing effectively avoids such arbitrages.

Fairness-aware Machine Learning (FairML) applications are often characterized by complex social objectives and legal requirements, frequently involving multiple, potentially conflicting notions of fairness. Despite the well-known Impossibility Theorem of Fairness and extensive theoretical research on the statistical and socio-technical trade-offs between fairness metrics, many FairML tools still optimize or constrain for a single fairness objective. However, this one-sided optimization can inadvertently lead to violations of other relevant notions of fairness. In this socio-technical and empirical study, we frame fairness as a many-objective (MaO) problem by treating fairness metrics as conflicting objectives. We introduce ManyFairHPO, a human-in-the-loop, fairness-aware model selection framework that enables practitioners to effectively navigate complex and nuanced fairness objective landscapes. ManyFairHPO aids in the identification, evaluation, and balancing of fairness metric conflicts and their related social consequences, leading to more informed and socially responsible model-selection decisions. Through a comprehensive empirical evaluation and a case study on the Law School Admissions problem, we demonstrate the effectiveness of ManyFairHPO in balancing multiple fairness objectives, mitigating risks such as self-fulfilling prophecies, and providing interpretable insights to guide stakeholders in making fairness-aware modeling decisions.

Fairness-aware Machine Learning (FairML) applications are often characterized by complex social objectives and legal requirements, frequently involving multiple, potentially conflicting notions of fairness. Despite the well-known Impossibility Theorem of Fairness and extensive theoretical research on the statistical and socio-technical trade-offs between fairness metrics, many FairML tools still optimize or constrain for a single fairness objective. However, this one-sided optimization can inadvertently lead to violations of other relevant notions of fairness. In this socio-technical and empirical study, we frame fairness as a Many-Objective (MaO) problem by treating fairness metrics as conflicting objectives in a multi-objective (MO) sense. We introduce ManyFairHPO, a human-in-the-loop, fairness-aware model selection framework that enables practitioners to effectively navigate complex and nuanced fairness objective landscapes. ManyFairHPO aids in the identification, evaluation, and balancing of fairness metric conflicts and their related social consequences, leading to more informed and socially responsible model-selection decisions. Through a comprehensive empirical evaluation and a case study on the Law School Admissions problem, we demonstrate the effectiveness of ManyFairHPO in balancing multiple fairness objectives, mitigating risks such as self-fulfilling prophecies, and providing interpretable insights to guide stakeholders in making fairness-aware modeling decisions.