Narrow Results

Let X be a stationary multifractal measure in the unit cube S₀ of R^d and X_r = X(S₀/r), r ≥ 1, be the measure in a smaller cube. We develop methods to calculate the marginal distributions of X_r and the coefficients C_j,i of the Haar wavelet representation of X. The methods are exact for discrete cascades and approximate for continuous-scaling multifractal measures. In the small scale limit r → ∞, the distribution of log[X_r] or log[X_r|X_r > 0] converges under renormalization to one of three different distribution types: normal, stable non-normal, or non-stable of a particular type. The normal distribution has the largest domain of attraction. The distribution of the absolute wavelet coefficients |C_j,i| at level j has similar asymptotic properties as j → ∞. We also develop a way to directly obtain the integer moments of X_r and C_j,i without preliminary calculation of the marginal distributions. Using the marginal distribution of the measure density ε_r = r^dX_r, we assess the accuracy for finite r of the asymptotic scaling relation P[ε_r > r^γ] ~ r^-c(γ), where c(γ) is the co-dimension function of X.

Straightforward methods to evaluate risks arising from several sources are specially difficult when risk components are dependent and, even more if that dependence is strong in the tails. We give an explicit analytical expression for the probability distribution of the sum of non-negative losses that are tail-dependent. Our model allows dependence in the extremes of the marginal beta distributions. The proposed model is flexible in the choice of the parameters in the marginal distribution. The estimation using the method of moments is possible and the calculation of risk measures is easily done with a Monte Carlo approach. An illustration on data for insurance losses is presented.

Climate change will push the weather experienced by people affected outside the bounds of historic norms, resulting in unprecedented weather events. But people and firms should be able to learn from their experience of unusual weather and adjust their expectations about the climate distribution accordingly. The efficiency of this learning process gives an upper bound on the rate at which adaptation can occur and is therefore important in determining the adjustment costs associated with climate change. Learning about climate change requires people to infer the state of a changing probability distribution (climate) given annual draws from that distribution (weather). If the climate is stationary, it can be inferred from the distribution of historic weather observations, but if it is changing, the inference problem is more challenging. This paper first develops different learning models, including an efficient hierarchical Bayesian model in which the observer learns whether the climate is changing and, if it is, the functional form that describes that change. I contrast this with a less efficient but simpler learning model in which observers react to past changes but are unable to anticipate future changes. I propose a general metric of learning costs based on the average, discounted squared difference between beliefs and the true climate state and use climate model output to calculate this metric for two emissions scenarios, finding substantial relative differences between learning models and scenarios but small absolute values. Geographic differences arise from spatial patterns of warming rates and natural weather variability (noise). Finally, I present results from an experimental game simulating the adaptation decision, which suggests that people are able to learn about a trending climate and respond proactively.

Future risks linked to extreme events and options for managing them are receiving increasing attention in the research and policy arena, where uncertainty is considered to be one of the most challenging aspects in regard to disasters triggered by natural hazards. To shed more light on this issue, this study conducts a detailed uncertainty assessment of a forward-looking country level catastrophe risk model for extreme flood events in Bangladesh and identifies how various sources of uncertainties contribute to the variability in modeling results. Alternative assumptions of climate, exposure and vulnerability parameters show that scenario uncertainty regarding socio-economic development — and exposure in particular — seems to dominate other sources of uncertainty. Importantly, this trend is particularly notable for the estimate of extreme events rather than annual average losses and for the prediction over the longer-time horizons rather than near future. It is concluded that there is ample need to better understand how future vulnerability and exposure will develop as they found to be core determinants of risk, apart from climate change, for increases in extreme losses. One viable way is the incorporation of bottom-up assessment of exposure asset build-up and further analysis of vulnerability drivers, which could reduce epistemic uncertainty regarding projection of catastrophic economic losses into future. It is suggested that the concept of “iterative risk management” may provide a feasible way to achieve reduction of these uncertainties in a step-by-step basis.

Extreme temperatures and high levels of air pollutants pose major risks to human health. Because heat waves and pollution episodes share common underlying drivers, they often coincide; additionally, they may worsen under future climate change and so it is extremely important to understand the extent of their co-occurrence so that proper mitigation and adaptation steps can be taken to minimize their impact. Here, we use 15 years of observations of temperature, and measurements of surface ozone and particulate matter (the two air pollutants with the greatest human health impact) to characterize the overlap of extreme pollution episodes and heat waves. We show that all three extremes occur in large-scale, multi-day, overlapping and coherent structures. Additionally, we find that the largest, longest lasting episodes have the greatest frequency of co-occurring extremes and have the highest temperatures and most extreme levels of pollution.

We review shock models for defaults, both in the standard and in the urn-based approach. Shock models are motivated by engineering problems where a material can break because of stress, but they can also be efficiently used in other fields, such as economics and biology.

Standard shock models are parametric models, whereas the urn-based shock models are nonparametric models assuming as little as possible for the prediction of defaults. First, we mention some results for the two models, in particular describing the finite and asymptotic behavior of time to failure or of the chance for defaults. Finally, we also present an application of the second model to defaults of Italian firms, comparing our results with a standard prediction model of economics, the Z-score. We note that our model predicts the default behavior of these firms better.