Tychastic Viability
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Tychastic Viability A Mathematical Approach to Time and Uncertainty Jean-Pierre Aubin
Received: 19 November 2012 / Accepted: 23 July 2013 / Published online: 4 September 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract Tychastic viability is defined in an uncertain dynamical framework and used for providing a ‘‘viability risk eradication measure’’, first, by delineating the set of initial conditions from which all evolutions satisfy viability constraints, second, for the other ‘‘risky’’ initial states, by introducing their duration index. This approach provides an alternative to the stochastic representation of chance and these two measures replace the statistical measures (expectation, variance, etc). Keywords
Viability Tychastic Insurance Stochastic
1 Introduction This survey provides a short account of ‘‘tychastic viability’’ and its application to define a concept of measure of risk of eradication of the viability of an environment under an ‘‘uncertain’’ dynamical system. Some of the results surveyed are scattered in the literature, and all of them will be presented in detail in the forthcoming monograph [5, Eradication of Risk] by L. Chen, O. Dordan and the author for financial applications, in [4, Time and Money] for economic ones and, among other ones, in for climate applications. Some domains of biology (see for instance [Demomgeot, 16 ] and [23, Murray]), cognitive sciences, for instance, have
The author is grateful to Luxi Chen, Giuseppe Da Prato, Olivier Dordan, Halim Doss and H. Frankowska for their recent and older cooperation on ‘‘tychastic approach’’ to uncertainty. This work was partially supported by the Commission of the European Communities under the 7th Framework Programme Marie Curie Initial Training Network (FP7-PEOPLE-2010-ITN), project SADCO, contract number 264735. J.-P. Aubin (&) VIMADES, 14, rue Domat, 75005 Paris, France e-mail: [email protected] URL: http://vimades.com/aubin/
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Fig. 1 Inequality constraints. For example if ‘ = 1, dynamical inequality constraints 8 t 2 R; xðtÞ kðtÞ ð2Þ are viability constraints: introducing the subsets K(t) : = { x such that x C k(t)}, they are equivalent to (1): 8 t 2 R; xðtÞ 2 KðtÞ
Fig. 2 Viable evolutions. Example of trajectories of two evolutions viable in the tube K, and thus, satisfying inequalities xi(t) C k(t), i = 1, 2. Evolutionary systems (evolutionary engines, so to speak) must provide evolutions which are all viable in a tubular environment
motivated viability problems, as well as homeostasis and morphogenesis ([17,18, Fronville]). Illustrations are provided in Figs. 1, 5, and 7 (Figs. 2, 3, 4, 6). Uncertainty can be made precise according to the question asked, either ‘‘average behavior’’, as in stochastic systems, or ‘‘worst case behavior’’, as in tychastic systems which are defined in this survey. Tychasticity is related to viability constraints and uncertain evolutionary systems. It cannot be compared to statistics, which deals with other measures of risk. However, there are links betwee
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