Lower Bounds of Martingale Measure Densities in the Dalang-morton-willinger Theorem

For a d-dimensional stochastic process (Sn) N n=0 we obtain criteria for the existence of an equivalent martingale measure, whose density z, up to a 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 ∈ L; f, z ∈ L, 1/p + 1/q = 1, where p ∈ [1,∞], and the case of N -period model for p = ∞. The mentioned criteria are expressed in terms of the conditional distributions of the increments of S, as well as in terms of the boundedness from above of an utility function related to some optimal investment problem under the loss constraints. Several examples are presented.