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Abstract
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In this talk, we shall undertake a study of random times on filtered probability spaces. One of the main messages is that, as long as distributional properties of adapted processes up to the random time are involved, there is no loss of generality in assuming that the random time has a very specific construction related to Cox processes. This perspective sheds an intuitive light on results in the theory of progressive enlargement of filtrations, as is the semimartingale decomposition result of Jeulin and Yor. Furthermore, one obtains a complete characterization of honest times which avoid all stopping times as last maxima of local martingales with continuous supremum, vanishing at infinity. Applications of the previous theory include distributional properties of one-dimensional transient diffusions up to certain random times, as well as the role of the numeraire portfolio in stochastic finance as an indicator of overall market performance. Further potential applications in optimal stopping and expected utility maximization from terminal wealth with a random time-horizon will be discussed. |
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