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

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

2005-04-22

arxiv.org/abs/math/0504462v1

Mathematics

Probability

Scientific paper

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.

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