A complete characterization of local martingales which are functions of Brownian motion and its maximum

Abstract

We prove the max-martingale conjecture of Obłój and Yor. We show that for a continuous local martingale $(N_t:t\ge 0)$ and a function $H:R\times R_{+}\to R$, $H(N_t,{sup}_{s\le t}{N}_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)={\int }_0^{y}f\left(s\right)ds-f\left(y\right)(x-y)+H(0,0)$. This readily implies, via Lévy's equivalence theorem, an analogous result with the maximum process replaced by the local time at $0$.