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Sticky processes, local and true martingales

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  • Mikl'os R'asonyi
  • Hasanjan Sayit

Abstract

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.

Suggested Citation

  • Mikl'os R'asonyi & Hasanjan Sayit, 2015. "Sticky processes, local and true martingales," Papers 1509.08280, arXiv.org, revised Mar 2017.
    • Handle: RePEc:arx:papers:1509.08280
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    References listed on IDEAS

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    1. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    2. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
    3. Attila Herczegh & Vilmos Prokaj & Mikl'os R'asonyi, 2013. "Diversity and no arbitrage," Papers 1301.4173, arXiv.org, revised Aug 2014.
    4. Paolo Guasoni & Mikl'os R'asonyi & Walter Schachermayer, 2008. "Consistent price systems and face-lifting pricing under transaction costs," Papers 0803.4416, arXiv.org.
    5. Paolo Guasoni & Miklós Rásonyi, 2015. "Fragility of arbitrage and bubbles in local martingale diffusion models," Finance and Stochastics, Springer, vol. 19(2), pages 215-231, April.
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    Cited by:

    1. Nicolas Hérault & Dean Hyslop & Stephen P. Jenkins & Roger Wilkins, 2024. "Rising top‐income persistence in Australia: Evidence from income tax data," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 70(1), pages 154-186, March.
    2. Huy N. Chau & Mikl'os R'asonyi, 2016. "Skorohod's representation theorem and optimal strategies for markets with frictions," Papers 1606.07311, arXiv.org, revised Apr 2017.

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