Families of update rules for non-additive measures: Applications in pricing risks

Wang et al. [Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics 21 (1997) 173–183] propose axioms for prices in an insurance market. Chateauneuf et al. [Choquet pricing for financial markets with frictions, Mathematical Finance 6 (1996) 323–330] propose similar axioms for prices in a financial market with frictions. As a result of these axioms, market prices can be represented by the Choquet integral with respect to a non-additive measure. In both insurance and financial pricing, it is important to update prices in light of newly available information. This updating can be achieved by conditioning the underlying non-additive measure. Denneberg [Conditioning (updating) non-additive measures, Annals of Operations Research 52 (1994) 21–42] studies three conditioning rules for updating non-additive measures. Two of these update rules, the Bayesʹ and the Dempster-Shafer, are extreme cases of a family of update rules, [Gilboa, Schmeidler, Updating ambiguous beliefs, Journal of Economic Theory 59 (1993) 33–49]. In this paper, we introduce a family of update rules more general than the one of Gilboa and Schmeidler. We also show how to embed the general and Dempster-Shafer update formulas in another family of update rules. We examine the properties of these two families of update rules and the resulting conditional prices.