Lower and upper bounds of martingale measure densities in continuous time markets

Abstract

In a continuous time market model we consider the problem of existence of an equivalent martingale measure with density lying within given lower and upper bounds and we characterize a necessary and sufficient condition for this. In this sense our main result can be regarded as a version of the fundamental theorem of asset pricing. In our approach we suggest an axiomatic description of prices on $L_p$-spaces (with $p\in [1,\infty)$) and we rely on extension theorems for operators.