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

2008/04/10 by Dmitry B. Rokhlin, Rokhlin, Dmitry B. · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60G42 #91B24 #91B28 #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60G42 #msc:91B24 #msc:91B28

paper · pdf · doi:10.48550/arxiv.0804.1761

19 pages

arxiv created 2008/04/10 · openalex publication_date 2008/04/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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∈ L^p; f,z∈ L^q, 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.

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• A proof of the Dalang-Morton-Willinger theorem

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