A quantitative and a dual version of the Halmos-Savage theorem with applications to mathematical finance

Abstract

The celebrated theorem of Halmos and Savage implies that if M is a set of $\mathbb{P}$-absolutely continuous probability measures Q on $(\Omega, \mathscr{F}, \mathbb{P})$ such that each $A \in \mathscr{F}, \mathbb{P}(A) > 0$ is charged by some $Q\in M$, that is, $Q(A) > 0$ (where the choice of Q depends on the set A), then -- provided M is closed under countable convex combinations -- we can find $Q_0 \in M$ with full support; that is, $\mathbb{P}(A) > 0$ implies $Q_0(A) > 0 $. We show a quantitative version: if we assume that, for $\varepsilon > 0$ and $\delta > 0$ fixed, $\mathbb{P}(A)> \varepsilon$ implies that there is $Q \in M$ and $Q(A) > \delta$, then there is $Q_0 \in M$ such that $\mathbb{P}(A) >4 \varepsilon$ implies $Q_0(A)>\varepsilon^2 \delta/2$. This version of the Halmos-Savage theorem also allows a "dualization" which we also prove in a quantitative and qualitative version. We give applications to asymptoic problems arising in mathematical finance and pertaining to the relation of the concept of "no arbitrage" and the existence of equivalent martingale measures for a sequence of stochastic processes.