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.
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Notes
- 1.
The theory presented here explains also Jump-Diffusion processes (Chichilnisky 2012), the existence of ‘heavy tails’ in power law distributions, and the lumpiness of most of the physical systems that we observe and measure.
- 2.
ϕ U (x) = 1 when x ∈ U and ϕ U (x) = 0 when x ∉ U.
- 3.
In this article for simplicity we make no difference between probabilities and relative likelihoods.
- 4.
This is Savage’s (1972) definition of probability, see also Kadane and O’Hagan (1995).
- 5.
Savage’s probabilities can be either purely finitely additive or countably additive. In that sense they include all the probabilities in this article. However this article will exclude probabilities that are either purely finitely additive, or those that are countably additive, requiring elements of both, and therefore our characterization of a probability is strictly finer than that Savage’s (1954), and different from the view of a measure as a countably additive set function in [15].
- 6.
An equivalent definition of Monotone Continuity is that for every two events E 1 and E 2 in {E α } = 1, 2… with W(E 1) > W(E 2), there exists N such that altering arbitrarily the events E 1 and E 2 on a subset E i, where i > N, does not alter the probability or relative likelihood ranking of the events, namely W(E 1 ′) > W(E 2 ′) where E 1 ′ and E 2 ′ are the altered events.
- 7.
See [15].
- 8.
Here E c denotes the complement of the set E, A′ = A a.e. on \(E_{j}^{c} \Leftrightarrow A' \cap E_{j}^{c} = A \cap E_{j}^{c}\) a.e.
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Acknowledgements
This article is an expression of gratitude to the memory of Jerry Marsden, a great mathematician and a wonderful man. As his first PhD student in pure Mathematics when he was a professor at the Mathematics Department of UC Berkeley, the author is indebted to Jerry Marsden for counseling and support in obtaining the first of her two PhDs at UC Berkeley, in pure Mathematics. The second PhD in Economics at UC Berkeley was obtained by the author with the counseling of the Nobel Laureate economist, Gerard Debreu. Jerry Marsden was critical to encourage the growth of the research in this article on new and more realistic axiomatic foundations of probability theory; he invited the author to organize a Workshop on Catastrophic Risks at the Fields Institute in 1996 where this research was introduced, and strongly encouraged since 1996 the continuation and growth of this research.
The author is Director, Columbia Consortium for Risk Management (CCRM) Columbia University, and Professor of Economics and of Mathematical Statistics, Columbia University, New York 10027, 335 Riverside Drive, NY 10025, tel. 212 678 1148, chichilnisky@columbia.edu; website: www.chichilnisky.com. We acknowledge support from Grant No 5222-72 of the US Air Force Office of Research directed by Professor Jun Zheng, Washington DC from 2009 to 2012. Initial results on Sustainable Development were presented at Stanford University’s 1993 Seminar on Reconsideration of Values, organized by Kenneth Arrow, at the National Bureau of Economic Research Conference Mathematical Economics: The Legacy of Gerard Debreu at UC Berkeley, October 21, 2005, the Department of Economics of the University of Kansas National Bureau of Economic Research General Equilibrium Conference, September 2006, at the Departments of Statistics of the University of Oslo, Norway, Fall 2007, at a seminar organized by the former Professor Christopher Heyde at the Department of Statistics of Columbia University, Fall 2007, at seminars organized by Drs. Charles Figuieres and Mabel Tidball at LAMETA Universite de Montpellier, France December 19 and 20, 2008, and by Professor Alan Kirman at GREQAM Universite de Marseille, December 18 2008. In December 8 2012, the work presented here and its applications were presented at an invited Plenary Key Note Presentation by the author to the Annual Meetings of the Canadian Mathematical Society Montreal Canada, December 8 2012. The work presented in this article is also the subject of a forthcoming Plenary Key Note Presentation to the Annual Meeting of the Australian Mathematical Society in Sidney, Australia, December 18, 2013. We are grateful to the above institutions and individuals for supporting the research, and for helpful comments and suggestions.
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Appendix
Appendix
1.1 Example: A Probability that is Biased Against Frequent Events
Consider W(f) = liminf x ε R (f(x)). This is insensitive to frequent events of arbitrarily large Lebesgue measure [18] and therefore does not satisfy Axiom 2. In addition it is not linear, failing Axiom 1.
1.2 Example: The Dual Space L ∞ ∗ Consists of Countably Additive and Finitely Additive Measures
The space of continuous linear functions on L ∞ is the ‘dual’ of L ∞ , and is denoted L ∞ ∗. It has been characterized e.g. in Yosida [27, 28]. L ∞ ∗ consists of the sum of two subspaces (i)L 1 functions g that define countably additive measures ν on R by the rule ν(A) = ∫ A g(x)d x where ∫ R ∣g(x)∣d x < ∞ so that υ is absolutely continuous with respect to the Lebesgue measure, and (i i) a subspace consisting of purely finitely additive measure. A countable measure can be identified with an L 1 function, called its ‘density,’ but purely finitely additive measures cannot be identified by such functions.
1.3 Example: A Finitely Additive Measure that is Not Countably Additive
See Example in Section 7.
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Chichilnisky, G. (2015). The Topology of Change: Foundations of Probability with Black Swans. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_4
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