A proof of the Dalang-Morton-Willinger theorem

2008/04/21 by Dmitry B. Rokhlin, Rokhlin, Dmitry B.
Economics, Econometrics and Finance · Mathematics · Social Sciences · #60G42 #91B24 #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60G42 #msc:91B24

paper · pdf · doi:10.48550/arxiv.0804.3308

9 pages

arxiv created 2008/04/21 · openalex publication_date 2008/04/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a new proof of the Dalang-Morton-Willinger theorem, relating the no-arbitrage condition in stochastic securities market models to the existence of an equivalent martingale measure with bounded density for a d-dimensional stochastic sequence (S_n)_n=0^N of stock prices. Roughly speaking, the proof is reduced to the assertion that under the no-arbitrage condition for N=1 and S∈ L¹ there exists a strictly positive linear fucntional on L¹, which is bounded from above on a special subset of the subspace K⊂ L¹ of investor's gains.

Citations

• Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem
• Kreps-Yan theorem for Banach ideal spaces

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