DUALITY AND GENERAL EQUILIBRIUM THEORY UNDER KNIGHTIAN UNCERTAINTY *

Patrick Beissner (1) , Laurent Denis (2)

1 University of Canberra
2 Laboratoire Manceau de Mathématiques

Any dynamic or stochastic notion of a general equilibrium relies on the underlying commodity space. Under sole risk and without multiple–prior uncertainty the usual choice is a Lebesgue space from standard measure theory. In the case of volatility uncertainty it turns out that such a type of function space is no longer appropriate. For this reason we introduce and discuss a new natural commodity space, which can be constructed in three independent and equivalent ways. Each approach departs from one possible way to construct Lebesgue spaces. Moreover, we give a complete representation of the resulting topological dual space. This extends the classic Riesz representation in a natural way. Elements therein are the candidates for a linear equilibrium price system. This representation result has direct implications for the microeconomic foundation of finance under Knightian uncertainty.

Mots clés: Asset pricing; general equilibrium under uncertainty; space of contingent claims; volatility uncertainty; dual space; mutually singular probability measures

Domaines: Probabilités [math.PR]

Patrick Beissner, Laurent Denis. DUALITY AND GENERAL EQUILIBRIUM THEORY UNDER KNIGHTIAN UNCERTAINTY *. SIAM Journal on Financial Mathematics, 2018, ⟨10.1137/17M1120877⟩. ⟨hal-01585973⟩