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Convex Analysisの二,三の進展について

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The value-at-risk (VaR) and the conditional value-at-risk (CVaR) are two commonly used risk measures. We state some of their properties and make a comparison. Moreover, the structure of the portfolio optimization problem using the VaR and CVaR objective is studied.

Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk

This paper introduces a set of axioms that define convex risk measures. Duality theory provides the representation theorem for these measures and the link with pricing rules.

Putting order in risk measures

In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability (Yaari, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

Axiomatic characterization of insurance prices

Annuitization and asset allocation

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

We introduce a class of partial orderings of risks that are dual to stochastic dominance orderings. These arise as “distortion-free” orderings in Yaari's dual theory of risk (1987). We show that these dual orderings are equivalent to inverse stochastic dominance orderings (Muliere and Scarsini, 1989). We motivate third dual stochastic dominance via insurance economics, while providing an alternative interpretation for second (dual) stochastic dominance. We apply dual stochastic dominance to actuarial science and show how the dual ordering of risks is related to ordering income distributions in the economics of income inequality.

Virginia R. Young

Papers

Axiomatic characterization of insurance prices

Annuitization and asset allocation

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Ordering risks: Expected utility theory versus Yaari's dual theory of risk

Fuzzy subsethood