We adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Although Africa is home to about 14% of the global population (1.14 billion people), it is growing three times faster than the global average [1]. The continent carries a high burden of disease, but there has been real progress in eradication, elimination, and control since 2015. Examples are the eradication of wild polio in 2020 [2] and the eradication or elimination of neglected tropical diseases, such as dracunculiasis in Kenya in 2018; Human African trypanosomiasis in Togo in 2022; and trachoma in Togo, Gambia, Ghana, and Malawi in 2022 [3]. New HIV infections reduced by 44% in 2021 compared to 2010 [4], and in 2021 the African region passed the 2020 milestone of the End TB Strategy, with a 22% reduction in new infections compared with 2015 [5].

We define and study properties of implied volatility for American perpetual put options. In particular, we show that if the market prices are derived from a local volatility model with a monotone volatility function, then the corresponding implied volatility is also monotone as a function of the strike price.

We analysed the transmission of the human mpox virus in Spain by estimating the effective reproduction number of the disease from official surveillance data. Our computations show that this decreased steadily after an initial burst phase, dropping below 1 on July 12, and thus the outbreak was expected to reduce in the following weeks. Differences in trends were found across geographical regions of the country and across MSM and heterosexual populations.

This paper investigates properties of the class of graphs based on exchangeable point processes. We provide asymptotic expressions for the number of edges, number of nodes, and degree distributions, identifying four regimes: (i) a dense regime, (ii) a sparse, almost dense regime, (iii) a sparse regime with power-law behaviour, and (iv) an almost extremely sparse regime. We show that, under mild assumptions, both the global and local clustering coefficients converge to constants which may or may not be the same. We also derive a central limit theorem for subgraph counts and for the number of nodes. Finally, we propose a class of models within this framework where one can separately control the latent structure and the global sparsity/power-law properties of the graph.

This work explores the use of Trusted Research Environments for the secure analysis of sensitive, record-level data on local coronavirus disease-2019 (COVID-19) inequalities and economic vulnerabilities. The Local Data Spaces (LDS) project was a targeted rapid response and cross-disciplinary collaborative initiative using the Office for National Statistics’ Secure Research Service for localized comparison and analysis of health and economic outcomes over the course of the COVID-19 pandemic. Embedded researchers worked on co-producing a range of locally focused insights and reports built on secure secondary data and made appropriately open and available to the public and all local stakeholders for wider use. With secure infrastructure and overall data governance practices in place, accredited researchers were able to access a wealth of detailed data and resources to facilitate more targeted local policy analysis. Working with data within such infrastructure as part of a larger research project involved advanced planning and coordination to be efficient. As new and novel granular data resources become securely available (e.g., record-level administrative digital health records or consumer data), a range of local policy insights can be gained across issues of public health or local economic vitality. Many of these new forms of data however often come with a large degree of sensitivity around issues of personal identifiability and how the data is used for public-facing research and require secure and responsible use. Learning to work appropriately with secure data and research environments can open up many avenues for collaboration and analysis.