The Topology of Change: Foundations of Probability with Black Swans

Abstract

Classic probability theory treats rare events as ‘outliers’ that are disregarded and underestimated. In a moment of change however rare events can become frequent, and frequent events rare. We postulate new axioms for probability theory that require a balanced treatment of rare and frequent events, based on what we call “the topology of change”. The axioms extend the foundation of probability to integrate rare but potentially catastrophic events or black swans: natural hazards, market crashes, catastrophic climate change and episodes of species extinction. The new results include a characterization of a family of purely finitely additive measures that are—somewhat surprisingly—absolutely continuous with respect to the Lebesgue measure. This is a new development from an earlier characterization of probability measures implied by the new axioms, which where countably additive measures created in Chichilnisky (2000), Wiley, Chichester (2002), Chichilnisky (2009, 2009a). The results are contrasted to the axioms of Kolmogorov (1933/1950), De Groot (1970/2004), Arrow (1971), Dubins and Savage (1965), Savage (1972), Von Neumann and Morgernstern (1944), and Hernstein and Milnor.