Jump to main content
Mathematical Finance
Research seminar
Mathematical Finance 

seminar: Quantitative Finance · Statistics · Related Areas

Time: Thursday 09:15

Organizers: Vladimir Shikhman, Alois Pichler

Room: RH 39/ 633 (C46.633)

Title: A branch-and-bound algorithm for non-convex Nash equilibrium problems

Abstract: We present the first spatial branch-and-bound method for the computation of the set of all epsilon-Nash equilibria of continuous box-constrained non-convex Nash equilibrium problems with an approximation guarantee. Thereby, the existence of epsilon-Nash equilibria is not assumed, but the algorithm is also able to detect their absence. After a brief introduction to branch-and-bound ideas in global optimization, we explain appropriate discarding and fathoming techniques for Nash equilibrium problems, formulate convergence results for the proposed algorithm, and report our computational experience.

Website of the speaker

Title: Stochastic Optimal Control of Heating Systems with a Geothermal Energy Storage

Abstract: We consider the cost-optimal management of a residential heating system equipped with several heat production and consumption units. The manager is exposed to uncertainties about randomly fluctuating renewable heat production and environmental conditions driving the heat demand and supply. As a special feature the manager has access to a geothermal storage which allows for intertemporal transfer of thermal energy. This leads to a challenging mathematical optimization problem. The optimization problem is treated as a continuous-time stochastic optimal control problem for a controlled state process whose dynamics is described by a system of ordinary differential equations (ODEs), stochastic differential equations (SDEs) and a partial differential equation (PDE). We first apply semi-discretization to the PDE and use model order reduction techniques to reduce the dimension of the associated system of ODEs. Our numerical experiments for the model reduction with the balanced truncation method show that the space-time dynamics of the temperature in the geothermal storage can be described by only a few controlled ODEs. Finally, time-discretization leads to a Markov decision process for which we apply numerical methods to determine a cost-optimal control and the associated value function.

Website of the speaker

Title: Mortality in Germany during the COVID-19 pandemic

Website of the speaker

Title: A new diversity measure with application on SARS-CoV-2 Variants

Abstract: We define a diversity measure based on a solid theoretical foundation, which gives us a framework that allows studying patterns of diversity among sub-populations. The term sub-populations can be interpreted at different levels of granularity, hence maximizing the potential applications of the method. Our approach allows studying diversity patterns through a straightforward graphical analysis. We apply the method to study: (i) the diversity of SARS-CoV-2 variants between the seven most modernised countries in the world (G7) and sub-Saharan African countries; (ii) the diversity of variants over a time period in some targeted countries, e.g., UK, USA, South Africa; and (iii) the diversity of the variants between sub-Saharan African countries, South Asian countries, and South American countries.

Title: Levenberg Marquardt: general convergence, stochastic case, constrained case for scientific computing

Website of the speaker

Title: Stochastic Control: Monge-Kantorovich problem (Wasserstein distance) to Schrödinger bridge problem

Website of the speaker

Title: Scaled cuts for stochastic mixed-integer programs

Abstract: We develop a new type of Benders’ decomposition for two-stage stochastic mixed-integer programs with general mixed-integer variables in both time stages. In this algorithm we iteratively construct tighter lower bounds of the expected second-stage cost function using a new family of so-called scaled optimality cuts. We derive these cuts by parametrically solving extended formulations of the second-stage problems using deterministic mixed-integer programming techniques. The advantage of these scaled cuts is that they allow for parametric non-linear feasibility cuts in the second stage, but that the optimality cuts in the master problem remain linear. We establish convergence by proving that the optimality cuts recover the convex envelope of the expected second-stage cost function.

Website of the speaker

Title: Margin calls and fire sales in low growth scenarios: Analysis via an agent-based stock market model

Abstract: Agent-based models are a helpful tool to analyze the occurrence of contagion dynamics and asset price crashes in financial markets. By building an agent-based stock market model with margin trading and conducting numerical simulations, we investigate whether lower growth rate regimes of fundamental values can have an impact on the stability of stock price dynamics. Traders are modeled as individual agents interacting on a stock market in which the market price is an emergent property of the decisions of traders. Financial stability is defined as the absence of large asset price crashes caused by overlapping portfolio contagion. We match a stylized fact from empirical data on margin trading and show the results of numerical simulations of the model.

Website of the speaker

Title: Robustness of stochastic programs with endogenous randomness via contamination

Abstract: Investigating stability of stochastic programs with respect to changes in the underlying probability distributions represents an important step before deploying any model to production. Often, the uncertainty in stochastic programs is not perfectly known, thus it is approximated. The stochastic distribution’s misspecification and approximation errors can affect model solution, consequently leading to suboptimal decisions. It is of utmost importance to be aware of such errors and to have an estimate of their influence on the model solution. One approach, which estimates the possible impact of such errors, is the contamination technique. The methodology studies the effect of perturbation in the probability distribution by some contaminating distribution on the optimal value of stochastic programs. Lower and upper bounds, for the optimal values of perturbed stochastic programs, have been developed for numerous types of stochastic programs with exogenous randomness. In this paper, we first extend the current results by developing a tighter lower bound applicable to wider range of problems. Thereafter, we define contamination for decision-dependent randomness stochastic programs and prove various lower and upper bounds. We split the various cases into two separate sub-classes based on whether the feasibility set is fixed or decision-dependent and discuss several tractable formulations. Finally, we illustrate the contamination results on a real example of a stochastic program with endogenous randomness from a financial industry.

Website of the speaker

Title: Estimating multiplicity of infection, allele frequencies, and prevalences accounting for incomplete data

Abstract: Background: Molecular surveillance of infectious diseases allows the mon itoring of pathogens beyond the granularity of traditional epidemiological approaches and is well-established for some of the most relevant infectious diseases such as malaria. The presence of genetically distinct pathogenic variants within an infection, referred to as multiplicity of infection (MOI) or complexity of infection (COI) is common in malaria and similar infectious diseases. It is an important metric that scales with transmission intensities, potentially affects the clinical pathogenesis, and a confounding factor when monitoring the frequency and prevalence of pathogenic variants. Several statistical methods exist to estimate MOI and the frequency distribution of pathogen variants. However, a common problem is the quality of the underlying molecular data. If molecular assays fail not randomly, it is likely to underestimate MOI and the prevalence of pathogen variants.
Methods and findings: A statistical model is introduced which explicitly addresses data quality, by assuming a probability by which a pathogen variant remains undetected in a molecular assay. This is different from the assumption of missing at random, for which a molecular assay either performs perfectly or fails completely. The method is applicable to a single molecular marker and allows to estimate allele-frequency spectra, the distribution of MOI, and the probability of variants to remain undetected (incomplete information). Based on the statistical model, expressions for the prevalence of pathogen variants are derived and differences between frequency and prevalence are discussed. The usual desirable asymptotic properties of the maximum-likelihood estimator (MLE) are established by rewriting the model into an exponential family. The MLE has promising finite sample properties in terms of bias and variance. The covariance matrix of the estimator is close to the Cramér-Rao lower bound (inverse Fisher information). Importantly, the estimator’s variance is larger than that of a similar method which disregards incomplete information, but its bias is smaller.
Conclusions: Although the model introduced here has convenient properties, in terms of the mean squared error it does not outperform a simple standard method that neglects missing information. Thus, the new method is recommendable only for data sets in which the molecular assays produced poor quality results. This will be particularly true if the model is extended to accommodate information from multiple molecular markers at the same time, and incomplete information at one or more markers leads to strong depletion of sample size.

Title: A maximum-likelihood method to estimate haplotype frequencies and prevalence alongside multiplicity of infection from SNP data

Abstract: Despite a tremendous improvement in control and prevention of malaria, it still represents a major epidemiological treat for numerous countries across the globe, especially in sub- Saharan Africa, where it is also an obstacle to a sustainable economic development. While several control programs were put into place to enhance surveillance, control, and eradication of the disease, reliable clinical and epidemiological data is often unavailable, and typical epidemiological quantities are notoriously difficult to estimate. The introduction of genomic/molecular methods allowed to access data with a more refined granularity than it had been possible with traditional methods, and hence better estimates. This is particularly true in disease surveillance, for the monitoring of phenomenon such as the spread of drug resistance. Unfortunately, such an endeavor is challenged by the complexity of the malaria transmission cycle, and the presence of multiple genetically distinct pathogen variants within the same infection. This is known as multiplicity of infection (MOI). Ad-hoc approaches that disregard the ambiguous information caused by MOI, are usually used for the estimation. However, this leads to less confident and biased estimates. To account for ambiguous information in the data, a concise statistical framework is required. Here, we introduce a statistical framework to obtain maximum-likelihood estimates (MLE) of haplotype frequencies and prevalence alongside MOI in the case of malaria, using SNP data, i.e., multiple biallelic marker loci. Here, the number of model parameters increases geometrically with the number of genetic markers considered and no closed-form solution exists for the MLE. We, therefore, derive the MLE numerically, by using the Expectation-Maximization (EM) algorithm. This algorithm turns out to be in our case, an efficient and easy-to-implement algorithm that yields a numerically stable solution. The performance of the derived estimator is assessed by a systematic numerical simulation. For reasonable sample sizes, and number of loci, the method has little bias and variance. To exemplify the use of the method, we apply it to a dataset from Cameroon on sulfadoxine-pyrimethamine resistance in Plasmodium falciparum malaria.

Title: Resistance to antimalarials in Plasmodium falciparum spreads if mutations increase parasite survival when encountering the drug. Because such mutations are believed to spread only if drug pressure is high, and drug sensitivity returns after drug use is discontinued. However, this was not observed with sulfadoxine-pyrimethamine (SP) resistance. When SP was discontinued in Peru the resistant 50R/51I/108N replaced the highly resistant 51I/108N/164L Pfdhfr variant. Prevalence data of SP resistant Pfdhfr mutations from Peru and results from published yeast expression systems for antifolate sensitivity are combined with an evolutionary-genetic statistical model tailored towards malaria to explain the spread of the 51I/108N/164L and 50R/51I/108N mutants. Drug pressure is necessary for the 50R/51I/108N mutant to emerge. However, it cannot become predominant in the presence of more resistant variants until drug pressure is lifted. Afterwards, the 50R/51I/108N mutant spreads and prevents the re-emergence of sensitive types.

Title: Levenberg Marquardt: general convergence, stochastic case, constrained case for scientific computing

Website of the speaker

Title: Comparison of Risk measures and relations to stochastic dominance

Abstract: Classical risk measures can be associated with a Banach space (specifically, a norm) and conversely, norms can be used to build risk measures. This talk relates selected risk measures and demonstrates how stochastic dominance relates to norms of general risk measures. Among these risk measures, expectiles exhibit unique properties which can be used in risk-averse regression.

Website of the speaker

Title: Optimal Strategies of ISO and Producers on Electricity Markets with Elastic Demand, Production Bounds and Costs

Website of the speaker

Title: Optimal Market Making under Model Uncertainty: A Robust Reinforcement Learning Approach

Abstract: We study the optimal market making problem in order-driven electronic markets, with a focus on model uncertainty. We consider ambiguity in order arrival intensities and derive a robust strategy that can perform under various market conditions. To achieve this, we introduce a tractable model for the limit order book using Markov Decision Processes and develop robust Reinforcement Learning to solve the complex optimization problem. This approach enables us to accurately represent the order book dynamics with tick structures, as opposed to the usual price dynamics modeled in stochastic approaches. This is a joint work with Hoang Hai Tran, Julian Sester and Yijiong Zhang.

Website of the speaker

Title: Entropy-Regularized Quantization in Risk Management: Theoretical Insights and Practical Applications

Abstract: In this presentation, we will thoroughly explore the theoretical foundations and real-world applications of our recent research on entropy-regularized quantization. Our novel approach is designed to approximate probability measures on $\mathbb{R}^d$ using finite and discrete measures, with the Wasserstein distance as the key metric for quality assessment. We will discuss the properties and robustness of this method, which relaxes the traditional quantization problem and utilizes the well-established softmin function known for its theoretical and practical reliability. Our implementation leverages a stochastic gradient approach to achieve optimal solutions, with a flexible control parameter to adapt to the difficulty levels of specific optimization problems. To illustrate the practical relevance of our research, we will provide a concrete application in the context of Risk Management. The presentation will not only highlight our theoretical findings but also showcase the method's empirical performance in real-world risk assessment scenarios.

Website of the speaker

Title: On local uniqueness of normalized Nash equilibria

Abstract: For generalized Nash equilibrium problems (GNEP) with shared constraints we focus on the notion of normalized Nash equilibrium in the nonconvex setting. The property of nondegeneracy for normalized Nash equilibria is introduced. Nondegeneracy refers to GNEP-tailored versions of linear independence constraint qualification, strict complementarity and second-order regularity. Surprisingly enough, nondegeneracy of normalized Nash equilibrium does not prevent from degeneracies at the individual players' level. We show that generically all normalized Nash equilibria are nondegenerate. Moreover, nondegeneracy turns out to be a sufficient condition for the local uniqueness of normalized Nash equilibria. We emphasize that even in the convex setting the proposed notion of nondegeneracy differs from the sufficient condition for (global) uniqueness of normalized Nash equilibria, which is known from the literature.

Website of the speaker

Title: Statistical methods to estimate haplotype frequencies and multiplicity of infection for molecular disease surveillance

Abstract: Disease surveillance is crucial for informed decision-making in public health. Genome sequencing techniques provide high-resolution data, enabling the detection and characterization of pathogen variants, i.e., haplotypes at the gene level. However, these methods yield ambiguous data, particularly in infections involving multiple genetically different pathogens. This phenomenon is known as multiplicity of infection (MOI), notably observed in diseases such as malaria, one of the world's deadliest infectious diseases. Ad-hoc methods are often used to estimate haplotype frequencies alongside MOI from molecular data. In these methods, ambiguous samples are typically discarded, yielding deflated sample sizes and significantly biased estimates. Various statistical methods have been developed to take advantage of ambiguous samples when estimating haplotype frequencies and MOI. However, they suffer from several limitations, i.e., restricted genetic architecture, unclear method formulation, and insufficiently understood finite sample properties. Here, we introduce a novel statistical framework to obtain maximum-likelihood estimates (MLEs) for the estimation of haplotype frequencies alongside MOI for theoretically any random genetic architecture. Due to the high dimensionality of the problem and the complexity of the likelihood function, the estimates are obtained numerically, using an expectation-maximization (EM) algorithm. Furthermore, we explicitly derive from the method a framework for the estimation of prevalence, a clinically relevant metric. A systematic numerical study that explores the performance of the method for small sample sizes demonstrates that the method has low bias, statistical efficiency, and the MLEs can be used reliably to derive quantities such as linkage disequilibrium. To illustrate the applicability of the method, it is applied to a molecular dataset of Plasmodium falciparum-positive blood samples related to Sulfadoxine-Pyrimethamine (SP) resistance. Note that, while the model is built with a focus on malaria, its utility extends to diseases with similar transmission patterns. An implementation of the model is provided as a ready-to-use R script available on GitHub.