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

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

2008-04-10

arxiv.org/abs/0804.1761v1

Mathematics

Probability

19 pages

Scientific paper

For a $d$-dimensional stochastic process $(S_n)_{n=0}^N$ 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 $S\in L^p$; $f,z\in L^q$, $1/p+1/q=1$, where $p\in [1,\infty]$, and the case of $N$-period model for $p=\infty$. The mentioned criteria are expressed in 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.

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