The Uniform Integrability of Martingales. On a Question by Alexander Cherny

Let $X$ be a progressively measurable, almost surely right-continuous stochastic process such that $X_\tau \in L^1$ and $E[X_\tau] = E[X_0]$ for each finite stopping time $\tau$. In 2006, Cherny showed that $X$ is then a uniformly integrable martingale provided that $X$ is additionally nonnegative. Cherny then posed the question whether this implication also holds even if $X$ is not necessarily nonnegative. We provide an example that illustrates that this implication is wrong, in general. If, however, an additional integrability assumption is made on the limit inferior of $|X|$ then the implication holds. Finally, we argue that this integrability assumption holds if the stopping times are allowed to be randomized in a suitable sense.Comment: Revised version. Accepted for publication in Stochastic Processes and their Application

The Martingale Property in the Context of Stochastic Differential Equations

This note studies the martingale property of a nonnegative, continuous local martingale Z, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition states that Z is a (uniformly integrable) martingale if and only if an integral test of a related functional holds.Comment: Revised version. Published in Electron. Commun. Proba

Discrete stochastic approximations of the Mumford-Shah functional

We propose a $\Gamma$-convergent discrete approximation of the Mumford-Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford-Shah functional in any dimension.Comment: 47 pages, reorganized versio

Conditioned Martingales

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.Comment: Corrected several typos, improved formulations. Accepted by Electronic Communications in Probability; Electronic Communications in Probability, 2012, Volume 17, Issue 4