arXiv subjects

Jan Obloj

Publications and source records attributed to Jan Obloj.

38 records · Page 3Linked to original sources

arXiv2005

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

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.

math.PR↗

arXiv2004

On local martingale and its supremum: harmonic functions and beyond

We discuss certain facts involving a continuous local martingale $N$ and its supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic functions is proposed. This yields an important family of martingales, the usefulness of which is demonstrated, by means of examples involving the Skorokhod embedding problem, bounds on the law of the supremum, or the local time at 0, of a martingale with a fixed terminal distribution, or yet in some Brownian penalization problems. In particular we obtain new bounds on the law of the local time at 0, which involve the excess wealth order.

math.PR↗