The Topology of Change Foundations of Probability with Black Swans Dedicated to the Memory of Jerrold Marsden

Abstract

Classic probability theory treats rare events as ‘outliers’ that are often disregarded and underestimated. Yet in a moment of change rare events can become frequent and frequent events rare. We therefore postulate new axioms for probability theory that require a balanced treatment for 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 major episodes of species extinction. The new results presented in this article 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 all the probabilities measures implied by the new axioms as a combination of purely finitely additive and countably additive measures that was first established in Chichilnisky (2000, 2002, 2009b) and the results are contrasted here to the work of Kolmogorov (1950), De Groot (1970), Arrow (1971), Dubins and Savage (1965), Savage (1972), Von Neumann and Morgernstern (1944), and Herstein and Milnor (1953).