# INVARIANCE TIMES *

Abstract : 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 $(\mathbb{F},\mathbb{Q})$ Az\'ema 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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https://hal.archives-ouvertes.fr/hal-01455414
Contributor : Stéphane Crépey <>
Submitted on : Friday, February 3, 2017 - 3:11:36 PM
Last modification on : Friday, July 20, 2018 - 11:13:03 AM
Long-term archiving on : Friday, May 5, 2017 - 11:40:53 AM

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• HAL Id : hal-01455414, version 1
• ARXIV : 1702.01045

### Citation

Stéphane Crépey, Shiqi Song. INVARIANCE TIMES *. 2017. ⟨hal-01455414⟩

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