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The main purpose of the book is to show how a viscosity approach can be used to tackle control problems in insurance. The problems covered are the maximization of survival probability as well as the maximization of dividends in the classical collective risk model. The authors consider the possibility of controlling the risk process by reinsurance as well as by investments. They show that optimal value functions are characterized as either the unique or the smallest viscosity solution of the associated Hamilton-Jacobi-Bellman equation; they also study the structure of the optimal strategies and show how to find them. The viscosity approach was widely used in control problems related to mathematical finance but until quite recently it was not used to solve control problems related to actuarial mathematical science. This book is designed to familiarize the reader on how to use this approach. The intended audience is graduate students as well as researchers in this area.

Molecular characterization of colorectal cancer has helped us understand better the biology of the disease. However, previous efforts have yet to provide significant clinical value in order to be integrated into clinical practice for patients with early‐stage colon cancer (CC). The purpose of this study was to assess PD‐L1, GLUT‐1, e‐cadherin, MUC2, CDX2, and microsatellite instability (dMMR) and to propose a risk‐panel with prognostic capabilities. Biomarkers were immunohistochemically assessed through tissue microarrays in a cohort of 144 patients with stage II/III colon cancer. A biomarker panel consisting of PD‐L1, GLUT‐1, dMMR, and potentially CDX2 was constructed that divided patients into low, medium, and high risk of overall survival or disease-free survival (DFS) in equally sized groups. Compared with low‐risk patients, medium‐risk patients have almost twice the risk of death (HR = 2.10 (0.99–4.46), p = 0.054), while high‐risk patients have almost four times the risk (HR = 3.79 (1.77–8.11), p = 0.001). The multivariate goodness of fit was 0.756 and was correlated with Kaplan–Meier curves (p = 0.002). Consistent results were found for DFS. This study provides a critical basis for the future development of an immunohistochemical assessment capable of discerning early‐stage CC patients as a function of their prognosis. This tool may aid with treatment personalization in daily clinical practice and improve survival outcomes. ; This research was partly funded by European Union Regional Development Fund (17–20, RefBioII, Trans-Pyrenean cooperation network program 2016–2018, INTERREG-POCTEFA)