Felix Liebrich: Lyapunov, FrAchet, Marinacci: A tale of one (or many) probabilities
A crucial condition that allows to establish that two probability charges lambda-system agree is Marinacci’s Uniqueness Theorem (MUT). A countably additive probability charge Q and a convex-ranged probability charge P agree if there is a nontrivial event such that Q is constant on the generated P-equilikelihood class. Special cases of the MUT can be derived from Lyapunov’s Convexity Theorem and the sharp version of the Fréchet-Hoeffding bounds. A further alternative formulation of the assumption of the MUT is that the Q-expectation functional on simple random variables is locally P-invariant. It is this overlap between invariance and uniqueness that is the focus of the talk.
In a first part, we look at functionals defined on random variables that are globally invariant with respect to a reference probability P, i.e., its value only depends on the distribution of its argument under P. In contrast to most of the literature on the topic, we take a concrete functional as given and ask the uniqueness question: Can there be more than one such such reference probability? For wide classes of functionals, we shall leverage Lyapunov’s Convexity Theorem and see that this is not the case unless they are (i) constant, or (ii) more generally depend only on the essential supremum and essential infimum of the argument.
In the second part of the talk we return to the MUT and the recent, but significant extension provided by Svistula that dispenses with the countable additivity of Q. We expand and clarify the nature of the Svistula-MUT and uncover several equivalent conditions that enhance the applicability of said result. Thereafter, we tie the Svistula-MUT to a novel set of applications by extending to the case of finitely additive probabilities the sharp Fréchet-Hoeffding bounds. Building on this, we first obtain sharper results on the uniqueness of the reference probabilities of invariant capacities than in the first half of the talk. At last, we extend to the case of a finitely additive reference measure the Jouini-Schachermayer-Touzi-Svindland result on the automatic Fatou property of distribution-invariant risk measures.
The talk is partially based on joint work with Massimiliano Amarante (Montréal) and Cosimo Munari (Zürich).
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