A note on lower bounds of martingale measure densities

Abstract

For a given element $f\in L^1$ and a convex cone $C\subset L^{\infty }$, $C\cap L_{+}^{\infty }=\{0\}$ we give necessary and sufficient conditions for the existence of an element $g\ge f$ lying in the polar of $C$. This polar is taken in $(L^{\infty }{)}^{\ast }$ and in $L^1$. In the context of mathematical finance the main result concerns the existence of martingale measures, whose densities are bounded from below by prescribed random variable.

9 pages