The Annals of Applied Probability

On the stochastic behaviour of optional processes up to random times

Constantinos Kardaras

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In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin–Yor decomposition formula via Girsanov’s theorem is provided.

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Ann. Appl. Probab., Volume 25, Number 2 (2015), 429-464.

First available in Project Euclid: 19 February 2015

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Zentralblatt MATH identifier

Primary: 60G07: General theory of processes 60G44: Martingales with continuous parameter

Random times randomised stopping times times of maximum last passage times


Kardaras, Constantinos. On the stochastic behaviour of optional processes up to random times. Ann. Appl. Probab. 25 (2015), no. 2, 429--464. doi:10.1214/13-AAP976.

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