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Invariance times

Author

Listed:
  • St'ephane Cr'epey

    (LaMME)

  • Shiqi Song

    (LaMME)

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.

Suggested Citation

  • St'ephane Cr'epey & Shiqi Song, 2017. "Invariance times," Papers 1702.01045, arXiv.org.
    • Handle: RePEc:arx:papers:1702.01045
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    References listed on IDEAS

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    1. Bo, Lijun & Capponi, Agostino, 2015. "Counterparty risk for CDS: Default clustering effects," Journal of Banking & Finance, Elsevier, vol. 52(C), pages 29-42.
    2. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    3. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    4. Damiano Brigo & Agostino Capponi & Andrea Pallavicini, 2014. "Arbitrage-Free Bilateral Counterparty Risk Valuation Under Collateralization And Application To Credit Default Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 125-146, January.
    5. Crépey, Stéphane & Song, Shiqi, 2015. "BSDEs of counterparty risk," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3023-3052.
    6. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    7. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    8. P. Collin-Dufresne & R. Goldstein & J. Hugonnier, 2004. "A General Formula for Valuing Defaultable Securities," Econometrica, Econometric Society, vol. 72(5), pages 1377-1407, September.
    9. Travis Fisher & Sergio Pulido & Johannes Ruf, 2017. "Financial Models with Defaultable Numéraires," Working Papers hal-01240736, HAL.
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