Optional decomposition and Lagrange multipliers

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.