Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions

Abstract

It is proved that monetary utility functions that are commonotonic and time-consistent are conditional expectations. We also give additional results on atomless and conditionally atomless probability spaces. These notions describe that in a filtration, there are many new events at each time step.

Conditionally atomless extensions of sigma algebras

“…In this paper 1 we will work with a probability space equipped with three sigma algebras (Ω, F 0 ⊂ F 1 ⊂ F 2 , P). The sigma algebra F 0 is supposed to be trivial F 0 = {∅, Ω} whereas the sigma algebra F 2 is supposed to express innovations with respect to F 1 .…”

“…We will use the notation K for such a kernel. More precisely: the mapping K : Ω × F 2 → R + satisfies (1) For almost every ω ∈ Ω, the mapping K(ω, .) : F 2 → [0, 1] is a probability.…”

“…Part of the results were developed and used in my paper on commonotonicity, see [1] Date: First version March 2020, this version March 23, 2020. 1 This paper is to be seen as an exercise in measure theory. It will not be sent to a math.…”

“…The usual proof -presented in many probability courses -uses the Axiom of Choice (AC). A referee of [1] pointed out that for many people AC -or Zorn's lemma -is an extra assumption. To prove Sierpiński's theorem we only need the Axiom of Countable Dependent Choice, which is a countable form of the axiom of choice.…”

Convex increasing functionals on $C_b(X)$ spaces

“…The present paper is a generalisation of the author's paper [3] on monetary utility functions, satisfying (because of concavity) a downward continuity assumption. The author is grateful to Patrick Cheridito and Michael Kupper for insisting on looking at the more general context of monotonic convex functions.…”