Analysis of Fourier transform valuation formulas and applications

The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ

Hopf bifurcation in a gene regulatory network model:Molecular movement causes oscillations

Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper we analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a 1-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour and this is the case for our model, shown initially via computational simulations. In order to investigate this behaviour more deeply, we next solve our system using Green's functions and then undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. This has implications for transcription factors such as p53, NF-$\kappa$B and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer

Consensus-based recommendations for diagnosis and surgical management of cranioplasty and post-traumatic hydrocephalus from a European panel

Introduction: Planning cranioplasty (CPL) in patients with suspected or proven post-traumatic hydrocephalus (PTH) poses a significant management challenge due to a lack of clear guidance. Research question: This project aims to create a European document to improve adherence and adapt to local protocols based on available resources and national health systems. Methods: After a thorough non-systematic review, a steering committee (SC) formed a European expert panel (EP) for a two-round questionnaire using the Delphi method. The questionnaire employed a 9-point Likert scale to assess the appropriateness of statements inherent to two sections: “Diagnostic criteria for PTH” and “Surgical strategies for PTH and cranial reconstruction.” Results: The panel reached a consensus on 29 statements. In the “Diagnostic criteria for PTH” section, five statements were deemed “appropriate” (consensus 74.2−90.3 %), two were labeled “inappropriate,” and seven were marked as “uncertain.” In the “Surgical strategies for PTH and cranial reconstruction” section, four statements were considered “appropriate” (consensus 74.2−90.4 %), six were “inappropriate,” and five were “uncertain.” Discussion and conclusion: Planning a cranioplasty alongside hydrocephalus remains a significant challenge in neurosurgery. Our consensus conference suggests that, in patients with cranial decompression and suspected hydrocephalus, the most suitable diagnostic approach involves a combination of evolving clinical conditions and neuroradiological imaging. The recommended management sequence prioritizes cranial reconstruction, with the option of a ventriculoperitoneal shunt when needed, preferably with a programmable valve. We strongly recommend to adopt local protocols based on expert consensus, such as this, to guide patient care