  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
$H:R x R\_+\to R$, $H(N\_t,\sup\_{s\leq t}N\_s)$ is a local martingale if and
only if there exists a locally integrable function $f$ such that
$H(x,y)=\int\_0^y f(s)ds-f(y)(x-y)+H(0,0)$. This implies readily, via Levy's
equivalence theorem, an analogous result with the maximum process replaced by
the local time at 0.
