Seminar: Finanzmathematik · Statistik · verwandte Gebiete

Title: Mortality in Germany during the COVID-19 pandemic

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

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

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.

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.

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.

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.

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