A proof of the Dalang-Morton-Willinger theorem

TL;DR

This paper presents a new proof of the Dalang-Morton-Willinger theorem, linking no-arbitrage conditions in financial models to the existence of an equivalent martingale measure with bounded density.

Contribution

It offers a novel proof approach for the theorem, simplifying the connection between no-arbitrage and martingale measures in stochastic securities markets.

Findings

Established the existence of an equivalent martingale measure with bounded density under no-arbitrage.

Reduced the proof to a linear functional existence on $L^1$ space.

Provided a new perspective on the theorem's proof structure.

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 $S\in L^1$ there exists a strictly positive linear fucntional on $L^1$, which is bounded from above on a special subset of the subspace $K\subset L^1$ of investor's gains.