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Two Brownian particles with rank-based characteristics and skew-elastic collisions

Author

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  • Fernholz, E. Robert
  • Ichiba, Tomoyuki
  • Karatzas, Ioannis

Abstract

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems.

Suggested Citation

  • Fernholz, E. Robert & Ichiba, Tomoyuki & Karatzas, Ioannis, 2013. "Two Brownian particles with rank-based characteristics and skew-elastic collisions," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2999-3026.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:8:p:2999-3026
    DOI: 10.1016/j.spa.2013.03.019
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    References listed on IDEAS

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    1. Mitchel Y. Abolafia (ed.), 2005. "Markets," Books, Edward Elgar Publishing, number 2788, August.
    2. Petit, Frédérique, 1997. "Time reversal and reflected diffusions," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 25-53, July.
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    Cited by:

    1. Benjamin Jourdain & Julien Reygner, 2013. "Capital distribution and portfolio performance in the mean-field Atlas model," Papers 1312.5660, arXiv.org, revised Aug 2014.
    2. Antoine Lejay, 2018. "Estimation of the bias parameter of the skew random walk and application to the skew Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 539-551, October.
    3. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    4. Robert Fernholz, 2017. "Stratonovich representation of semimartingale rank processes," Papers 1705.00336, arXiv.org.
    5. Andrey Sarantsev, 2014. "On a class of diverse market models," Annals of Finance, Springer, vol. 10(2), pages 291-314, May.
    6. Peter Grandits & Maike Klein, 2020. "Ruin probability in a two-dimensional model with correlated Brownian motions," Papers 2004.13601, arXiv.org.
    7. Benjamin Jourdain & Julien Reygner, 2015. "Capital distribution and portfolio performance in the mean-field Atlas model," Post-Print hal-00921151, HAL.
    8. Alexander Vervuurt, 2015. "Topics in Stochastic Portfolio Theory," Papers 1504.02988, arXiv.org.
    9. Atar, Rami & Budhiraja, Amarjit, 2015. "On the multi-dimensional skew Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1911-1925.

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