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Probability measure-valued polynomial diffusions

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  • Christa Cuchiero
  • Martin Larsson
  • Sara Svaluto-Ferro

Abstract

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

Suggested Citation

  • Christa Cuchiero & Martin Larsson & Sara Svaluto-Ferro, 2018. "Probability measure-valued polynomial diffusions," Papers 1807.03229, arXiv.org.
    • Handle: RePEc:arx:papers:1807.03229
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    References listed on IDEAS

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    1. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    2. Ahdida, Abdelkoddousse & Alfonsi, Aurélien, 2013. "A mean-reverting SDE on correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1472-1520.
    3. Kurtz, Thomas G. & Xiong, Jie, 1999. "Particle representations for a class of nonlinear SPDEs," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 103-126, September.
    4. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    5. Abdelkoddousse Ahdida & Aurélien Alfonsi, 2013. "A Mean-Reverting SDE on Correlation matrices," Post-Print hal-00617111, HAL.
    6. Vaillancourt, Jean, 1990. "Interacting Fleming-Viot processes," Stochastic Processes and their Applications, Elsevier, vol. 36(1), pages 45-57, October.
    7. Boumezoued, Alexandre & Hardy, Héloïse Labit & El Karoui, Nicole & Arnold, Séverine, 2018. "Cause-of-death mortality: What can be learned from population dynamics?," Insurance: Mathematics and Economics, Elsevier, vol. 78(C), pages 301-315.
    8. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    9. Damir Filipović & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Swiss Finance Institute Research Paper Series 17-60, Swiss Finance Institute.
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