13,529 research outputs found
The Uniform Integrability of Martingales. On a Question by Alexander Cherny
Let be a progressively measurable, almost surely right-continuous
stochastic process such that and for each
finite stopping time . In 2006, Cherny showed that is then a
uniformly integrable martingale provided that is additionally nonnegative.
Cherny then posed the question whether this implication also holds even if
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 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 -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
A one-dimensional diffusion hits points fast
A one-dimensional, continuous, regular, and strong Markov process with
state space hits any point fast with positive probability. To
wit, if , then for all and
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