## The Annals of Applied Probability

### On the stochastic behaviour of optional processes up to random times

Constantinos Kardaras

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 19 February 2015

https://projecteuclid.org/euclid.aoap/1424355119

Digital Object Identifier
doi:10.1214/13-AAP976

Mathematical Reviews number (MathSciNet)
MR3313744

Zentralblatt MATH identifier
1316.60057

#### Citation

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. https://projecteuclid.org/euclid.aoap/1424355119

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