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For a $d$-dimensional stochastic process $(S_n)_{n=0}^N$ we obtain criteriafor the existence of an equivalent martingale measure, whose density $z$, up toa normalizing constant, is bounded from below by a given random variable $f$.We consider the case of one-period model (N=1) under the assumptions $S\inL^p$; $f,z\in L^q$, $1/p+1/q=1$, where $p\in [1,\infty]$, and the case of$N$-period model for $p=\infty$. The mentioned criteria are expressed in termsof the conditional distributions of the increments of $S$, as well as in termsof the boundedness from above of an utility function related to some optimalinvestment problem under the loss constraints. Several examples are presented.

Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem