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Separating Times for One-Dimensional Diffusions

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  • David Criens
  • Mikhail Urusov

Abstract

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.

Suggested Citation

  • David Criens & Mikhail Urusov, 2022. "Separating Times for One-Dimensional Diffusions," Papers 2211.06042, arXiv.org, revised May 2023.
    • Handle: RePEc:arx:papers:2211.06042
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    References listed on IDEAS

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    1. Prokaj, Vilmos & Bondici, László, 2022. "On the lack of semimartingale property," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 335-359.
    2. David Criens, 2018. "No Arbitrage in Continuous Financial Markets," Papers 1809.09588, arXiv.org, revised Feb 2020.
    3. Sascha Desmettre & Gunther Leobacher & L. C. G. Rogers, 2021. "Change of drift in one-dimensional diffusions," Finance and Stochastics, Springer, vol. 25(2), pages 359-381, April.
    4. Aleksandar Mijatović & Mikhail Urusov, 2012. "Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models," Finance and Stochastics, Springer, vol. 16(2), pages 225-247, April.
    5. Aleksandar Mijatović & Mikhail Urusov, 2015. "On the Loss of the Semimartingale Property at the Hitting Time of a Level," Journal of Theoretical Probability, Springer, vol. 28(3), pages 892-922, September.
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