The Omega model: from bankruptcy to occupation times in the red

Ruin occurs the first time when the surplus of a company or an institution is negative. In the Omega model, it is assumed that even with a negative surplus, the company can do business as usual until bankruptcy occurs. The probability of bankruptcy at a point of time only depends on the value of the negative surplus at that time. Under the assumption of Brownian motion for the surplus, the expected discounted value of a penalty at bankruptcy is determined, and hence the probability of bankruptcy. There is an intrinsic relation between the probability of no bankruptcy and an exposure random variable. In special cases, the distribution of the total time the Brownian motion spends below zero is found, and the Laplace transform of the integral of the negative part of the Brownian motion is expressed in terms of the Airy function of the first kind.

Natural vitamin C intake and the risk of head and neck cancer : a pooled analysis in the International Head and Neck Cancer Epidemiology Consortium

Evidence of associations between single nutrients and head and neck cancer (HNC) is still more limited and less consistent than that for fruit and vegetables. However, clarification of the protective mechanisms of fruit and vegetables is important to our understanding of HNC etiology. We investigated the association between vitamin C intake from natural sources and cancer of the oral cavity/pharynx and larynx using individual-level pooled data from ten case-control studies (5,959 cases and 12,248 controls) participating in the International Head and Neck Cancer Epidemiology (INHANCE) consortium. After harmonization of study-specific exposure information via the residual method, adjusted odds ratios (ORs) and corresponding 95% confidence intervals (CIs) were estimated using unconditional multiple logistic regression models on quintile categories of 'non-alcohol energy-adjusted' vitamin C intake. In the presence of heterogeneity of the estimated ORs among studies, we derived those estimates from generalized linear mixed models. Higher intakes of vitamin C were inversely related to oral and pharyngeal (OR = 0.54, 95% CI: 0.45-0.65, for the fifth quintile category versus the first one, p for trend<0.001) and laryngeal cancers (OR = 0.52, 95% CI: 0.40-0.68, p for trend = 0.006), although in the presence of heterogeneity among studies for both sites. Inverse associations were consistently observed for the anatomical subsites of oral and pharyngeal cancer, and across strata of age, sex, education, body mass index, tobacco, and alcohol, for both cancer sites. The inverse association of vitamin C intake from foods with HNC may reflect a protective effect on these cancers; however, we cannot rule out other explanations.