The paper defines and studies thin times which are random times whose graph is contained in a countable union of graphs of stopping times with respect to a reference filtration . We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all -stopping times. Then, for a given random time τ, we introduce , the smallest right-continuous filtration containing and making τ a stopping time, and we show that, for a thin time τ, each -martingale is an -semimartingale, i.e., the hypothesis for holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.
AA and MJ wish to acknowledge the generous financial supports of ‘Chaire Markets in transition’, French Banking Federation and ILB, Labex ANR 11-LABX-0019. AA wishes to acknowledge the support of the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 335421. The research of TC is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), through Grant RGPIN 04987. We thank an anonymous referee for useful comments.
"Thin times and random times’ decomposition." Electron. J. Probab. 26 1 - 22, 2021. https://doi.org/10.1214/20-EJP569