Optional decomposition and Lagrange multipliers

Abstract

Let \({\cal Q}\) be the set of equivalent martingale measures for a given process \(S\), and let \(X\) be a process which is a local supermartingale with respect to any measure in \({\cal Q}\). The optional decomposition theorem for \(X\) states that there exists a predictable integrand \(\varphi\) such that the difference \(X-\varphi\cdot S\) is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.