A stochastic model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic is considered. Infectives have independent and identically distributed infectivity profiles, which describe their infectiousness as a function of time since infection. The individual-to-individual infection rate depends also on the number of susceptibles present in the population. Exact results are derived for the distribution of statistics defined on the final outcome of the epidemic, including its final size. These are proved by using a generalisation of a Sellke construction to show that the distribution of the final outcome of the epidemic is the same as that of an associated discrete-time epidemic process, in which infectives are considered one at a time, and exploiting connection with death processes to analyse the final outcome of the latter. The results generalise easily to multipopulation epidemics.

In this paper we propose a new efficient algorithm to compute the value function for zero-sum stopping games featuring two players with opposing interests. This can be seen as a game version of the ‘forward algorithm’ for (one-player) optimal stopping problems, first introduced by Irle (2006) for discrete-time Markov chains and later revisited by Miclo and Villeneuve (2021) for continuous-time Markov processes on general state spaces. This paper focuses on a game driven by a homogeneous continuous-time Markov chain taking values in a finite state space and also discusses the number of iterations needed. Illustrated computational implementations for a few particular examples are also provided.

Federated Learning is a novel method of training machine learning models, pioneered by Google, aimed for use on smartphones. In contrast to traditional machine learning, where data is centralised and brought to the model, Federated Learning involves the algorithm being brought to the data, ensuring privacy is preserved. This paper will demonstrate how insurance companies in a market could use this technique to build a claims frequency neural network prediction model collectively by combining and using all of their customer data, without actually sharing or compromising any sensitive information with each other. A simulated car insurance market with 10 players was created using the freMTPL2freq dataset. It was found that if all insurers were permitted to share their confidential data with each other, they could collectively build a model that achieved 5.57% of exposure weighted Poisson Deviance Explained (% PDE) on an unseen sample. However, if they are not permitted to share their customer data, none of them can achieve more than 3.82% exposure weighted PDE on the same unseen sample. With Federated Learning, they can retain all of their customer data privately and construct a model that achieves a similar level of accuracy to that achieved by centralising all the data for model training, reaching 5.34% exposure weighted PDE on the same unseen sample.

We study a stochastic control problem where the underlying process follows a spectrally negative Lévy process. A controller can continuously increase the process but only decrease it at independent Poisson arrival times. We show the optimality of the periodic–classical barrier strategy, which increases the process whenever it would fall below some lower barrier and decreases it whenever it is observed above a higher barrier. An optimal strategy and the value function are written semi-explicitly using scale functions. Numerical results are also given.

Owing to their innovative guarantee features, the popularity of variable annuities has gained significant traction as suitable retirement products in recent years. Amongst these guarantees, the guaranteed minimum income benefit (GMIB) stands out as an appealing rider that can be integrated into variable annuity contracts. In this research, we construct a comprehensive modelling framework that encompasses three sources of uncertainty, namely interest risk, mortality risk and investment risk, with the aim of valuing the GMIB. These risk factors are modelled stochastically whilst accounting for the interdependence between interest and mortality risks. The numéraire transformation technique is utilised in our approach, capitalising on the concepts of the forward and endowment-risk-adjusted measures. By considering two distinct settings of the Benefit Base functions, we derive an analytic solution for the GMIB. Our numerical findings demonstrate the superiority of our proposed methodology vis-á-vis the standard Monte Carlo simulation as a benchmark in terms of computational accuracy and efficiency, achieving a remarkable average improvement of 99% computing time reduction compared to the benchmark. Furthermore, we conduct an extensive sensitivity analysis to explore the levels of impact of various model parameters on the value of the GMIB.

Lifetime pension pools—also known as group self-annuitization plans, pooled annuity funds, and variable payment life annuities in the literature—offer retirees lifelong income by collectively managing mortality risk and adjusting benefits based on the investment performance and the mortality experience within the pool. The benefit structure hinges on two key design parameters: the investment policy and the hurdle rate. However, past research offers limited guidance on optimal asset allocation in such settings, often relying on overly simplistic strategies. Furthermore, the choice of hurdle rate has received virtually no attention in the literature. This study addresses this gap by jointly analyzing optimal hurdle rates and investment strategies using a dynamic programming approach that allows for varying degrees of risk aversion via a hyperbolic absolute risk aversion utility function. Our findings reveal that, as risk aversion increases, the model favours more conservative portfolios and lower hurdle rates; conversely, lower risk aversion supports riskier allocations and higher hurdle rates. The threshold parameter—which reflects the minimum acceptable level of consumption—plays a critical role in shaping the hurdle rate behaviour.

We consider instrumental variables (IV) estimation of a possibly infinite order dynamic panel autoregressive (AR) process with individual effects. The estimation is based on the sieve AR approximation, with its lag order increasing with sample size. Transforming the variable to eliminate individual effects generates an endogeneity problem, particularly when the time series is only moderately long. IV approaches are useful to obtain well-behaved estimators in panels with large cross sections. We establish the consistency and asymptotic normality of the IV estimators, including the Anderson-Hsiao, generalized method of moments, and double filter IV (DFIV) estimators. The theoretical results are obtained under homoskedasticity using double asymptotics under which both the cross-sectional sample size and the length of the time series tend to infinity. The finite-sample performance of the estimators is examined using Monte Carlo simulation. Our preferred estimator is the DFIV estimator, as it exhibits excellent performance in terms of bias and coverage probability, despite its finite-sample distribution being relatively dispersed.

Core-periphery (CP) structure is frequently observed in networks where the nodes form two distinct groups: a small, densely interconnected core and a sparse periphery. Borgatti and Everett (Borgatti, S. P., & Everett M. G. (2000). Models of core/periphery structures. Social Networks, 21(4), 375–395.) proposed one of the most popular methods to identify and quantify CP structure by comparing the observed network with an “ideal” CP structure. While this metric has been widely used, an improved algorithm is still needed. In this work, we detail a greedy, label-switching algorithm to identify CP structure that is both fast and accurate. By leveraging a mathematical reformulation of the CP metric, our proposed heuristic offers an order-of-magnitude improvement on the number of operations compared to a naive implementation. We prove that the algorithm monotonically ascends to a local maximum while consistently yielding solutions within 90% of the global optimum on small toy networks. On synthetic networks, our algorithm exhibits superior classification accuracies and run-times compared to a popular competing method, and on one-real- world network, it is 340 times faster.

We study the multiserver-job setting in the load-focused multilevel scaling limit, where system load approaches capacity much faster than the growth of the number of servers $n$. We consider the “1 and $n$” system, where each job requires either one server or all $n$. Within the multilevel scaling limit, we examine three regimes: load dominated by $n$-server jobs, 1-server jobs, or balanced. In each regime, we characterize the asymptotic growth rate of the boundary of the stability region and the scaled mean queue length. We demonstrate that mean queue length peaks near balanced load via theory, numerics, and simulation.

Given a collection $\mathcal{D} =\{D_1,D_2,\ldots ,D_m\}$ of digraphs on the common vertex set $V$, an $m$-edge digraph $H$ with vertices in $V$ is transversal in $\mathcal{D}$ if there exists a bijection $\varphi \,:\,E(H)\rightarrow [m]$ such that $e \in E(D_{\varphi (e)})$ for all $e\in E(H)$. Ghouila-Houri proved that any $n$-vertex digraph with minimum semi-degree at least $\frac {n}{2}$ contains a directed Hamilton cycle. In this paper, we provide a transversal generalisation of Ghouila-Houri’s theorem, thereby solving a problem proposed by Chakraborti, Kim, Lee, and Seo. Our proof utilises the absorption method for transversals, the regularity method for digraph collections, as well as the transversal blow-up lemma and the related machinery. As an application, when $n$ is sufficiently large, our result implies the transversal version of Dirac’s theorem, which was proved by Joos and Kim.

Consider a general mortality-linked security (MLS) with a bounded payoff contingent on the evolution of the underlying mortality rate and the performance of associated risky assets. The mortality rate and asset prices are assumed to jointly follow a multivariate Itô process, driven by both a multivariate Brownian motion and a Poisson point process. We follow the utility indifference approach to pricing this MLS under the physical measure. To this end, we employ backward stochastic differential equations (BSDEs) to characterize the optimal investment strategy and the value function for the involved optimization problems. We then solve the resulting nonlinear BSDEs with a non-Lipschitz generator. This methodology, which combines the utility indifference approach with BSDE techniques, provides numerical tractability through Monte Carlo simulations. Finally, we conduct comprehensive numerical studies on the valuation of several concrete MLSs, with a focus on the sensitivity analysis of the indifference prices against various key model parameters, including, in particular, the correlation between the underlying mortality rate and asset price.

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.

Occupational blood exposure accidents (OBEAs) pose significant risks to healthcare workers, potentially exposing them to hepatitis B (HBV), hepatitis C (HCV), and HIV. While most research focuses on hospital settings, this study assessed OBEA management in non-hospital contexts. Although our data predate the COVID pandemic, findings remain highly relevant, especially for healthcare professionals working outside hospital settings. A retrospective analysis of OBEA registry data (2006–2014) was conducted in a southern Dutch region. Data included demographics, profession, workplace, injury type, source status (HBV, HCV, HIV), risk assessment, post-exposure measures, and lab results. Chi-square and t-tests were applied. In total, 975 OBEA were reported. Among nurses, medical assistants, students, and housekeeping staff, subcutaneous needles (51–67%) and lancets (25%) were common exposure sources. Police officers mainly reported biting (26%), scratching, or spitting (70%). HBV vaccination coverage ranged from 18% (housekeeping) to over 90% (nurses, police). Post-exposure measures were taken in 52% of cases. High-risk exposures (43%) mainly affected ambulance staff, sterilization workers, police, and dentists. Sources were tested in 85% of high-risk cases: 1.4% were HBV positive, 2% HCV positive, and 1.1% HIV positive. No seroconversions occurred. Results stress the need for better HBV vaccination coverage, targeted prevention, and prompt OBEA reporting outside hospital settings.

This study assessed changes in complete pneumococcal vaccination coverage (CPVC) among Peruvian children <5 years before and after the COVID 19 pandemic and evaluated regional differences, associated sociodemographic factors and wealth-related inequality. 2018–2023 Demographic and Health Surveys (DHS) was analyzed. CPVC was defined as receiving the full 2 + 1 schedule of the 13 valent pneumococcal vaccine. Children aged 13–60 months were included. Multivariable analysis used modified Poisson regression and wealth related inequality was assessed using the Concentration index and Erreygers’s corrected Concentration index at national and regional levels. Among 95,586 children, CPVC decreased from 71.9% in 2019 to 69.4% in 2020 (p = 0.003), then returned to pre Covid levels from 2021 onward (72.2% in 2023; p = 0.001), particularly in Lima Metropolitana. Puno (53.3–58.6%) and Madre de Dios (50.9–62.1%) consistently showed the lowest coverage. Nationally, wealth- or sociodemographic related inequalities were minimal; however, regional interactions indicated that the effect of wealth on CPVC varied by area. Depending on the region, factors such as age group, household members and mather’s education were associated with lower CPVC, whereas age at first pregnancy, institutional birth, antenatal care and access to information increased CPVC. Ucayali showed persistently higher CPVC among wealthier populations. Despite a temporary decline during the pandemic, CPVC in Peru rapidly recovered, although regional gaps persist.

We prove that for any $k\geq 3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the ‘replica symmetric solution’ predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. 76 (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq 3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)]. The improvement is significant particularly for small $k$.

Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon–Frankl–Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.

Counting the number of isomers of a chemical molecule is one of the formative problems of graph theory. However, recent progress has been slow, and the problem has largely been ignored in modern network science. Here we provide an introduction to the mathematics of counting network structures and then use it to derive results for two new classes of molecules. In contrast to previously studied examples, these classes take additional chemical complexity into account and thus require the use of multivariate generating functions. The results illustrate the elegance of counting theory, highlighting it as an important tool that should receive more attention in network science.

This paper presents an illustrated tutorial for conducting an embedded Mixed-Method Social Network Analysis (MMSNA) to examine the dynamic interplay between human agency and social networks. We draw on an empirical study in education that investigated how teachers enact relational agency within their school networks to support the integration of migrant students. We propose a replicable method and stepwise procedure for designing, implementing and evaluating an embedded MMSNA. While the potential of MMSNA has long been recognized across disciplines, its purpose and operationalization are often underexplained. We illustrate how MMSNA can be used to analyze both network structures and the agency of actors embedded within them, in alignment with specific research objectives and theoretical perspectives.

Fractional Brownian motion, with its long-time correlated increments, has been applied in many fields in recent years. Since volatility was shown to be rough by Gatheral, Jaisson, and Rosenbaum, fractional Brownian motion has gained popularity as a financial model. In this work, we revisit the definitions and properties of the univariate and multivariate fractional Brownian motions, and consider four simulation methods. We demonstrate the issues associated with applying the standard Euler scheme for simulating stochastic processes driven by fractional Brownian motion with $H < \frac{1}{2}$ (which we call the rough models). We then introduce a novel approximate method for simulating such rough models based on the fast algorithm by Ma and Wu, which accounts for a factor of 10 speedup. Finally, we consider applications of these methods to option pricing.