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

Abstract

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