Ruf, J;
(2015)
The Uniform Integrability of Martingales. On a Question by Alexander Cherny.
Stochastic Processes and their Applications
, 125
(10)
pp. 3657-3662.
10.1016/j.spa.2015.04.002.
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Abstract
Let X be a progressively measurable, almost surely right-continuous stochastic process such that Xτ∈L1 and E[Xτ]=E[X0] for each finite stopping time τ. 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.
| Type: | Article |
|---|---|
| Title: | The Uniform Integrability of Martingales. On a Question by Alexander Cherny |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.spa.2015.04.002 |
| Publisher version: | http://dx.doi.org/10.1016/j.spa.2015.04.002 |
| Language: | English |
| Additional information: | © 2015 Elsevier B.V. This manuscript is made available under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International license (CC BY-NC-ND 4.0). This license allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licenses are available at http://creativecommons.org/ licenses/by/4.0. Access may be initially restricted by the publisher. |
| Keywords: | Stopping time; Uniform integrability |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1466259 |
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