Open Access
November 2017 Invariance times
Stéphane Crépey, Shiqi Song
Ann. Probab. 45(6B): 4632-4674 (November 2017). DOI: 10.1214/17-AOP1174


On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$, we consider two filtrations $\mathbb{F}\subset\mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$ predictable processes on $(0,\theta]$. In this setup, it is well known that, for any $\mathbb{F}$ semimartingale $X$, the process $X^{\theta-}$ ($X$ stopped “right before $\theta$”) is a $\mathbb{G}$ semimartingale. Given a positive constant $T$, we call $\theta$ an invariance time if there exists a probability measure $\mathbb{P}$ equivalent to $\mathbb{Q}$ on $\mathcal{F}_{T}$ such that, for any $(\mathbb{F},\mathbb{P})$ local martingale $X$, $X^{\theta-}$ is a $(\mathbb{G},\mathbb{Q})$ local martingale. We characterize invariance times in terms of the $(\mathbb{F},\mathbb{Q})$ Azéma supermartingale of $\theta$ and we derive a mild and tractable invariance time sufficiency condition. We discuss invariance times in mathematical finance and BSDE applications.


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Stéphane Crépey. Shiqi Song. "Invariance times." Ann. Probab. 45 (6B) 4632 - 4674, November 2017.


Received: 1 September 2015; Revised: 1 July 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06838129
MathSciNet: MR3737920
Digital Object Identifier: 10.1214/17-AOP1174

Primary: 60G07
Secondary: 60G44

Keywords: Enlargement of filtration , mathematical finance , measure change , random time

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 6B • November 2017
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